Parameters: Inputs File

This documents on how to use WarpX with an inputs file (e.g., warpx.3d input_3d).

Complete example input files can be found in the examples section.

Note

WarpX input options are read via AMReX ParmParse.

Note

The AMReX parser (see Math parser and user-defined constants) is used for the right-hand-side of all input parameters that consist of one or more integers or floats, so expressions like <species_name>.density_max = "2.+1." and/or using user-defined constants are accepted.

Overall simulation parameters

• authors (string: e.g. "Jane Doe <jane@example.com>, Jimmy Joe <jimmy@example.com>")

  Authors of an input file / simulation setup. When provided, this information is added as metadata to (openPMD) output files.

• max_step (integer)

  The number of PIC cycles to perform.

• stop_time (float; in seconds)

  The maximum physical time of the simulation. Can be provided instead of max_step. If both max_step and stop_time are provided, both criteria are used and the simulation stops when the first criterion is hit.

  Note: in boosted-frame simulations, stop_time refers to the time in the boosted frame.

• warpx.used_inputs_file (string; default: warpx_used_inputs)

  Name of a file that WarpX writes to archive the used inputs. The context of this file will contain an exact copy of all explicitly and implicitly used inputs parameters, including those extended and overwritten from the command line.

• warpx.gamma_boost (float)

  The Lorentz factor of the boosted frame in which the simulation is run. (The corresponding Lorentz transformation is assumed to be along warpx.boost_direction.)

  When using this parameter, the input parameters are interpreted as in the lab-frame and automatically converted to the boosted frame. (See the corresponding documentation of each input parameters for exceptions.)

• warpx.boost_direction (string: x, y or z)

  The direction of the Lorentz-transform for boosted-frame simulations (The direction y cannot be used in 2D simulations.)

• warpx.zmax_plasma_to_compute_max_step (float) optional

  Can be useful when running in a boosted frame. If specified, automatically calculates the number of iterations required in the boosted frame for the lower z end of the simulation domain to reach warpx.zmax_plasma_to_compute_max_step (typically the plasma end, given in the lab frame). The value of max_step is overwritten, and printed to standard output. Currently only works if the Lorentz boost and the moving window are along the z direction.

• warpx.compute_max_step_from_btd (integer; 0 by default) optional

  Can be useful when computing back-transformed diagnostics. If specified, automatically calculates the number of iterations required in the boosted frame for all back-transformed diagnostics to be completed. If max_step, stop_time, or warpx.zmax_plasma_to_compute_max_step are not specified, or the current values of max_step and/or stop_time are too low to fill all BTD snapshots, the values of max_step and/or stop_time are overwritten with the new values and printed to standard output.

• warpx.random_seed (string or int > 0) optional

  If provided warpx.random_seed = random, the random seed will be determined using std::random_device and std::clock(), thus every simulation run produces different random numbers. If provided warpx.random_seed = n, and it is required that n > 0, the random seed for each MPI rank is (mpi_rank+1) * n, where mpi_rank starts from 0. n = 1 and warpx.random_seed = default produce the default random seed. Note that when GPU threading is used, one should not expect to obtain the same random numbers, even if a fixed warpx.random_seed is provided.

• algo.evolve_scheme (string, default: explicit)

  Specifies the evolve scheme used by WarpX.

  • explicit: Use an explicit solver, such as the standard FDTD or PSATD

  • implicit_picard: Use an implicit solver with exact energy conservation that uses a Picard iteration to solve the system. Note that this method is for demonstration only. It is inefficient and does not work well when \(\omega_{pe} \Delta t\) is close to or greater than one. The method is described in Angus et al., On numerical energy conservation for an implicit particle-in-cell method coupled with a binary Monte-Carlo algorithm for Coulomb collisions. The version implemented is an updated version that is relativistically correct, including the relativistic gamma factor for the particles. For exact energy conservation, algo.current_deposition = direct must be used with interpolation.galerkin_scheme = 0, and algo.current_deposition = Esirkepov must be used with interpolation.galerkin_scheme = 1 (which is the default, in which case charge will also be conserved).

  • semi_implicit_picard: Use an energy conserving semi-implicit solver that uses a Picard iteration to solve the system. Note that this method has the CFL limitation \(\Delta t < c/\sqrt( \sum_i 1/\Delta x_i^2 )\). It is inefficient and does not work well or at all when \(\omega_{pe} \Delta t\) is close to or greater than one. The method is described in Chen et al., A semi-implicit, energy- and charge-conserving particle-in-cell algorithm for the relativistic Vlasov-Maxwell equations. For energy conservation, algo.current_deposition = direct must be used with interpolation.galerkin_scheme = 0, and algo.current_deposition = Esirkepov must be used with interpolation.galerkin_scheme = 1 (which is the default, in which case charge will also be conserved).

• algo.max_picard_iterations (integer, default: 10)

  When algo.evolve_scheme is either implicit_picard or semi_implicit_picard, this sets the maximum number of Picard itearations that are done each time step.

• algo.picard_iteration_tolerance (float, default: 1.e-7)

  When algo.evolve_scheme is either implicit_picard or semi_implicit_picard, this sets the convergence tolerance of the iterations, the maximum of the relative change of the L2 norm of the field from one iteration to the next. If this is set to zero, the maximum number of iterations will always be done with the change only calculated on the last iteration (for a slight optimization).

• algo.require_picard_convergence (bool, default: 1)

  When algo.evolve_scheme is either implicit_picard or semi_implicit_picard, this sets whether the iteration each step is required to converge. If it is required, an abort is raised if it does not converge and the code then exits. If not, then a warning is issued and the calculation continues.

• warpx.do_electrostatic (string) optional (default none)

  Specifies the electrostatic mode. When turned on, instead of updating the fields at each iteration with the full Maxwell equations, the fields are recomputed at each iteration from the Poisson equation. There is no limitation on the timestep in this case, but electromagnetic effects (e.g. propagation of radiation, lasers, etc.) are not captured. There are several options:

  • labframe: Poisson’s equation is solved in the lab frame with the charge density of all species combined. More specifically, the code solves:

    \[\boldsymbol{\nabla}^2 \phi = - \rho/\epsilon_0 \qquad \boldsymbol{E} = - \boldsymbol{\nabla}\phi\]

  • labframe-electromagnetostatic: Poisson’s equation is solved in the lab frame with the charge density of all species combined. Additionally the 3-component vector potential is solved in the Coulomb Gauge with the current density of all species combined to include self magnetic fields. More specifically, the code solves:

    \[\begin{split}\boldsymbol{\nabla}^2 \phi = - \rho/\epsilon_0 \qquad \boldsymbol{E} = - \boldsymbol{\nabla}\phi \\ \boldsymbol{\nabla}^2 \boldsymbol{A} = - \mu_0 \boldsymbol{j} \qquad \boldsymbol{B} = \boldsymbol{\nabla}\times\boldsymbol{A}\end{split}\]

  • relativistic: Poisson’s equation is solved for each species in their respective rest frame. The corresponding field is mapped back to the simulation frame and will produce both E and B fields. More specifically, in the simulation frame, this is equivalent to solving for each species

    \[\boldsymbol{\nabla}^2 - (\boldsymbol{\beta}\cdot\boldsymbol{\nabla})^2\phi = - \rho/\epsilon_0 \qquad \boldsymbol{E} = -\boldsymbol{\nabla}\phi + \boldsymbol{\beta}(\boldsymbol{\beta} \cdot \boldsymbol{\nabla}\phi) \qquad \boldsymbol{B} = -\frac{1}{c}\boldsymbol{\beta}\times\boldsymbol{\nabla}\phi\]

    where \(\boldsymbol{\beta}\) is the average (normalized) velocity of the considered species (which can be relativistic). See, e.g., Vay [1] for more information.

• warpx.poisson_solver (string) optional (default multigrid)

  • multigrid: Poisson’s equation is solved using an iterative multigrid (MLMG) solver.

    See the AMReX documentation for details of the MLMG solver (the default solver used with electrostatic simulations). The default behavior of the code is to check whether there is non-zero charge density in the system and if so force the MLMG solver to use the solution max norm when checking convergence. If there is no charge density, the MLMG solver will switch to using the initial guess max norm error when evaluating convergence and an absolute error tolerance of \(10^{-6}\) \(\mathrm{V/m}^2\) will be used (unless a different non-zero value is specified by the user via warpx.self_fields_absolute_tolerance).

  • fft: Poisson’s equation is solved using an Integrated Green Function method (which requires FFT calculations).

    See these references for more details , . It only works in 3D and it requires the compilation flag -DWarpX_PSATD=ON. If mesh refinement is enabled, this solver only works on the coarsest level. On the refined patches, the Poisson equation is solved with the multigrid solver. In electrostatic mode, this solver requires open field boundary conditions (boundary.field_lo,hi = open). In electromagnetic mode, this solver can be used to initialize the species’ self fields (<species_name>.initialize_self_fields=1) provided that the field BCs are PML (boundary.field_lo,hi = PML).

• warpx.self_fields_required_precision (float, default: 1.e-11)

  The relative precision with which the electrostatic space-charge fields should be calculated. More specifically, the space-charge fields are computed with an iterative Multi-Level Multi-Grid (MLMG) solver. This solver can fail to reach the default precision within a reasonable time. This only applies when warpx.do_electrostatic = labframe.

• warpx.self_fields_absolute_tolerance (float, default: 0.0)

  The absolute tolerance with which the space-charge fields should be calculated in units of \(\mathrm{V/m}^2\). More specifically, the acceptable residual with which the solution can be considered converged. In general this should be left as the default, but in cases where the simulation state changes very little between steps it can occur that the initial guess for the MLMG solver is so close to the converged value that it fails to improve that solution sufficiently to reach the self_fields_required_precision value.

• warpx.self_fields_max_iters (integer, default: 200)

  Maximum number of iterations used for MLMG solver for space-charge fields calculation. In case if MLMG converges but fails to reach the desired self_fields_required_precision, this parameter may be increased. This only applies when warpx.do_electrostatic = labframe.

• warpx.self_fields_verbosity (integer, default: 2)

  The verbosity used for MLMG solver for space-charge fields calculation. Currently MLMG solver looks for verbosity levels from 0-5. A higher number results in more verbose output.

• amrex.abort_on_out_of_gpu_memory (0 or 1; default is 1 for true)

  When running on GPUs, memory that does not fit on the device will be automatically swapped to host memory when this option is set to 0. This will cause severe performance drops. Note that even with this set to 1 WarpX will not catch all out-of-memory events yet when operating close to maximum device memory. Please also see the documentation in AMReX.

• amrex.the_arena_is_managed (0 or 1; default is 0 for false)

  When running on GPUs, device memory that is accessed from the host will automatically be transferred with managed memory. This is useful for convenience during development, but has sometimes severe performance and memory footprint implications if relied on (and sometimes vendor bugs). For all regular WarpX operations, we therefore do explicit memory transfers without the need for managed memory and thus changed the AMReX default to false. Please also see the documentation in AMReX.

• amrex.omp_threads (system, nosmt or positive integer; default is nosmt)

  An integer number can be set in lieu of the OMP_NUM_THREADS environment variable to control the number of OpenMP threads to use for the OMP compute backend on CPUs. By default, we use the nosmt option, which overwrites the OpenMP default of spawning one thread per logical CPU core, and instead only spawns a number of threads equal to the number of physical CPU cores on the machine. If set, the environment variable OMP_NUM_THREADS takes precedence over system and nosmt, but not over integer numbers set in this option.

Signal Handling

WarpX can handle Unix (Linux/macOS) process signals. This can be useful to configure jobs on HPC and cloud systems to shut down cleanly when they are close to reaching their allocated walltime or to steer the simulation behavior interactively.

Allowed signal names are documented in the C++ standard and POSIX. We follow the same naming, but remove the SIG prefix, e.g., the WarpX signal configuration name for SIGINT is INT.

• warpx.break_signals (array of string, separated by spaces) optional

  A list of signal names or numbers that the simulation should handle by cleanly terminating at the next timestep

• warpx.checkpoint_signals (array of string, separated by spaces) optional

  A list of signal names or numbers that the simulation should handle by outputting a checkpoint at the next timestep. A diagnostic of type checkpoint must be configured.

Note

Certain signals are only available on specific platforms, please see the links above for details. Typically supported on Linux and macOS are HUP, INT, QUIT, ABRT, USR1, USR2, TERM, TSTP, URG, and IO among others.

Signals to think about twice before overwriting in interactive simulations: Note that INT (interupt) is the signal that Ctrl+C sends on the terminal, which most people use to abort a process; once overwritten you need to abort interactive jobs with, e.g., Ctrl+\ (QUIT) or sending the KILL signal. The TSTP (terminal stop) command is sent interactively from Ctrl+Z to temporarily send a process to sleep (until send in the background with commands such as bg or continued with fg), overwriting it would thus disable that functionality. The signals KILL and STOP cannot be used.

The FPE and ILL signals should not be overwritten in WarpX, as they are controlled by AMReX for debug workflows that catch invalid floating-point operations.

Tip

For example, the following logic can be added to Slurm batch scripts (signal name to number mapping here) to gracefully shut down 6 min prior to walltime. If you have a checkpoint diagnostics in your inputs file, this automatically will write a checkpoint due to the default <diag_name>.dump_last_timestep = 1 option in WarpX.

#SBATCH --signal=1@360

srun ...                   \
  warpx.break_signals=HUP  \
  > output.txt

For LSF batch systems, the equivalent job script lines are:

#BSUB -wa 'HUP' -wt '6'

jsrun ...                  \
  warpx.break_signals=HUP  \
  > output.txt

Setting up the field mesh

• amr.n_cell (2 integers in 2D, 3 integers in 3D)

  The number of grid points along each direction (on the coarsest level)

• amr.max_level (integer, default: 0)

  When using mesh refinement, the number of refinement levels that will be used.

  Use 0 in order to disable mesh refinement. Note: currently, 0 and 1 are supported.

• amr.ref_ratio (integer per refined level, default: 2)

  When using mesh refinement, this is the refinement ratio per level. With this option, all directions are fined by the same ratio.

• amr.ref_ratio_vect (3 integers for x,y,z per refined level)

  When using mesh refinement, this can be used to set the refinement ratio per direction and level, relative to the previous level.

  Example: for three levels, a value of 2 2 4 8 8 16 refines the first level by 2-fold in x and y and 4-fold in z compared to the coarsest level (level 0/mother grid); compared to the first level, the second level is refined 8-fold in x and y and 16-fold in z.

• geometry.dims (string)

  The dimensions of the simulation geometry. Supported values are 1, 2, 3, RZ. For 3, a cartesian geometry of x, y, z is modeled. For 2, the axes are x and z and all physics in y is assumed to be translation symmetric. For 1, the only axis is z and the dimensions x and y are translation symmetric. For RZ, we apply an azimuthal mode decomposition, with warpx.n_rz_azimuthal_modes providing further control.

  Note that this value has to match the WarpX_DIMS compile-time option. If you installed WarpX from a package manager, then pick the right executable by name.

• warpx.n_rz_azimuthal_modes (integer; 1 by default)

  When using the RZ version, this is the number of azimuthal modes. The default is 1, which corresponds to a perfectly axisymmetric simulation.

• geometry.prob_lo and geometry.prob_hi (2 floats in 2D, 3 floats in 3D; in meters)

  The extent of the full simulation box. This box is rectangular, and thus its extent is given here by the coordinates of the lower corner (geometry.prob_lo) and upper corner (geometry.prob_hi). The first axis of the coordinates is x (or r with cylindrical) and the last is z.

• warpx.do_moving_window (integer; 0 by default)

  Whether to use a moving window for the simulation

• warpx.moving_window_dir (either x, y or z)

  The direction of the moving window.

• warpx.moving_window_v (float)

  The speed of moving window, in units of the speed of light (i.e. use 1.0 for a moving window that moves exactly at the speed of light)

• warpx.start_moving_window_step (integer; 0 by default)

  The timestep at which the moving window starts.

• warpx.end_moving_window_step (integer; default is -1 for false)

  The timestep at which the moving window ends.

• warpx.fine_tag_lo and warpx.fine_tag_hi (2 floats in 2D, 3 floats in 3D; in meters) optional

  When using static mesh refinement with 1 level, the extent of the refined patch. This patch is rectangular, and thus its extent is given here by the coordinates of the lower corner (warpx.fine_tag_lo) and upper corner (warpx.fine_tag_hi).

• warpx.ref_patch_function(x,y,z) (string) optional

  A function of x, y, z that defines the extent of the refined patch when using static mesh refinement with amr.max_level>0. Note that the function can be used to define distinct regions for refinement, however, the refined regions should be such that the pml layer surrounding the patches should not overlap. For this reason, when defining distinct patches, please ensure that they are sufficiently separated.

• warpx.refine_plasma (integer) optional (default 0)

  Increase the number of macro-particles that are injected “ahead” of a mesh refinement patch in a moving window simulation.

  Note: in development; only works with static mesh-refinement, specific to moving window plasma injection, and requires a single refined level.

• warpx.n_current_deposition_buffer (integer)

  When using mesh refinement: the particles that are located inside a refinement patch, but within n_current_deposition_buffer cells of the edge of this patch, will deposit their charge and current to the lower refinement level, instead of depositing to the refinement patch itself. See the mesh-refinement section for more details. If this variable is not explicitly set in the input script, n_current_deposition_buffer is automatically set so as to be large enough to hold the particle shape, on the fine grid

• warpx.n_field_gather_buffer (integer, optional)

  Default: warpx.n_field_gather_buffer = n_current_deposition_buffer + 1 (one cell larger than n_current_deposition_buffer on the fine grid).

  When using mesh refinement, particles that are located inside a refinement patch, but within n_field_gather_buffer cells of the edge of the patch, gather the fields from the lower refinement level, instead of gathering the fields from the refinement patch itself. This avoids some of the spurious effects that can occur inside the refinement patch, close to its edge. See the section Mesh refinement for more details.

• warpx.do_single_precision_comms (integer; 0 by default)

  Perform MPI communications for field guard regions in single precision. Only meaningful for WarpX_PRECISION=DOUBLE.

• particles.deposit_on_main_grid (list of strings)

  When using mesh refinement: the particle species whose name are included in the list will deposit their charge/current directly on the main grid (i.e. the coarsest level), even if they are inside a refinement patch.

• particles.gather_from_main_grid (list of strings)

  When using mesh refinement: the particle species whose name are included in the list will gather their fields from the main grid (i.e. the coarsest level), even if they are inside a refinement patch.

Domain Boundary Conditions

• boundary.field_lo and boundary.field_hi (2 strings for 2D, 3 strings for 3D, pml by default)

  Boundary conditions applied to fields at the lower and upper domain boundaries. Options are:

  • Periodic: This option can be used to set periodic domain boundaries. Note that if the fields for lo in a certain dimension are set to periodic, then the corresponding upper boundary must also be set to periodic. If particle boundaries are not specified in the input file, then particles boundaries by default will be set to periodic. If particles boundaries are specified, then they must be set to periodic corresponding to the periodic field boundaries.

  • pml (default): This option can be used to add Perfectly Matched Layers (PML) around the simulation domain. See the PML theory section for more details. Additional pml algorithms can be explored using the parameters warpx.do_pml_in_domain, warpx.pml_has_particles, and warpx.do_pml_j_damping.

  • absorbing_silver_mueller: This option can be used to set the Silver-Mueller absorbing boundary conditions. These boundary conditions are simpler and less computationally expensive than the pml, but are also less effective at absorbing the field. They only work with the Yee Maxwell solver.

  • damped: This is the recommended option in the moving direction when using the spectral solver with moving window (currently only supported along z). This boundary condition applies a damping factor to the electric and magnetic fields in the outer half of the guard cells, using a sine squared profile. As the spectral solver is by nature periodic, the damping prevents fields from wrapping around to the other end of the domain when the periodicity is not desired. This boundary condition is only valid when using the spectral solver.

  • pec: This option can be used to set a Perfect Electric Conductor at the simulation boundary. Please see the PEC theory section for more details. Note that PEC boundary is invalid at r=0 for the RZ solver. Please use none option. This boundary condition does not work with the spectral solver.

  • none: No boundary condition is applied to the fields with the electromagnetic solver. This option must be used for the RZ-solver at r=0.

  • neumann: For the electrostatic multigrid solver, a Neumann boundary condition (with gradient of the potential equal to 0) will be applied on the specified boundary.

  • open: For the electrostatic Poisson solver based on a Integrated Green Function method.

• boundary.potential_lo_x/y/z and boundary.potential_hi_x/y/z (default 0)

  Gives the value of the electric potential at the boundaries, for pec boundaries. With electrostatic solvers (i.e., with warpx.do_electrostatic = ...), this is used in order to compute the potential in the simulation volume at each timestep. When using other solvers (e.g. Maxwell solver), setting these variables will trigger an electrostatic solve at t=0, to compute the initial electric field produced by the boundaries.

• boundary.particle_lo and boundary.particle_hi (2 strings for 2D, 3 strings for 3D, absorbing by default)

  Options are:

  • Absorbing: Particles leaving the boundary will be deleted.

  • Periodic: Particles leaving the boundary will re-enter from the opposite boundary. The field boundary condition must be consistently set to periodic and both lower and upper boundaries must be periodic.

  • Reflecting: Particles leaving the boundary are reflected from the boundary back into the domain. When boundary.reflect_all_velocities is false, the sign of only the normal velocity is changed, otherwise the sign of all velocities are changed.

  • Thermal: Particles leaving the boundary are reflected from the boundary back into the domain and their velocities are thermalized. The tangential velocity components are sampled from gaussian distribution and the component normal to the boundary is sampled from gaussian flux distribution. The standard deviation for these distributions should be provided for each species using boundary.<species>.u_th. The same standard deviation is used to sample all components.

• boundary.reflect_all_velocities (bool) optional (default false)

  For a reflecting boundary condition, this flags whether the sign of only the normal velocity is changed or all velocities.

• boundary.verboncoeur_axis_correction (bool) optional (default true)

  Whether to apply the Verboncoeur correction on the charge and current density on axis when using RZ. For nodal values (rho and Jz), the cell volume for values on axis is calculated as \(\pi*\Delta r^2/4\). In Verboncoeur [2], it is shown that using \(\pi*\Delta r^2/3\) instead will give a uniform density if the particle density is uniform.

Additional PML parameters

• warpx.pml_ncell (int; default: 10)

  The depth of the PML, in number of cells.

• do_similar_dm_pml (int; default: 1)

  Whether or not to use an amrex::DistributionMapping for the PML grids that is similar to the mother grids, meaning that the mapping will be computed to minimize the communication costs between the PML and the mother grids.

• warpx.pml_delta (int; default: 10)

  The characteristic depth, in number of cells, over which the absorption coefficients of the PML increases.

• warpx.do_pml_in_domain (int; default: 0)

  Whether to create the PML inside the simulation area or outside. If inside, it allows the user to propagate particles in PML and to use extended PML

• warpx.pml_has_particles (int; default: 0)

  Whether to propagate particles in PML or not. Can only be done if PML are in simulation domain, i.e. if warpx.do_pml_in_domain = 1.

• warpx.do_pml_j_damping (int; default: 0)

  Whether to damp current in PML. Can only be used if particles are propagated in PML, i.e. if warpx.pml_has_particles = 1.

• warpx.v_particle_pml (float; default: 1)

  When warpx.do_pml_j_damping = 1, the assumed velocity of the particles to be absorbed in the PML, in units of the speed of light c.

• warpx.do_pml_dive_cleaning (bool)

  Whether to use divergence cleaning for E in the PML region. The value must match warpx.do_pml_divb_cleaning (either both false or both true). This option seems to be necessary in order to avoid strong Nyquist instabilities in 3D simulations with the PSATD solver, open boundary conditions and PML in all directions. 2D simulations and 3D simulations with open boundary conditions and PML only in one direction might run well even without divergence cleaning. This option is implemented only for the Cartesian PSATD solver; it is turned on by default in this case.

• warpx.do_pml_divb_cleaning (bool)

  Whether to use divergence cleaning for B in the PML region. The value must match warpx.do_pml_dive_cleaning (either both false or both true). This option seems to be necessary in order to avoid strong Nyquist instabilities in 3D simulations with the PSATD solver, open boundary conditions and PML in all directions. 2D simulations and 3D simulations with open boundary conditions and PML only in one direction might run well even without divergence cleaning. This option is implemented only for the Cartesian PSATD solver; it is turned on by default in this case.

Embedded Boundary Conditions

• warpx.eb_implicit_function (string)

  A function of x, y, z that defines the surface of the embedded boundary. That surface lies where the function value is 0 ; the physics simulation area is where the function value is negative ; the interior of the embeddded boundary is where the function value is positive.

• warpx.eb_potential(x,y,z,t) (string)

  Gives the value of the electric potential at the surface of the embedded boundary, as a function of x, y, z and t. With electrostatic solvers (i.e., with warpx.do_electrostatic = ...), this is used in order to compute the potential in the simulation volume at each timestep. When using other solvers (e.g. Maxwell solver), setting this variable will trigger an electrostatic solve at t=0, to compute the initial electric field produced by the boundaries. Note that this function is also evaluated inside the embedded boundary. For this reason, it is important to define this function in such a way that it is constant inside the embedded boundary.

Distribution across MPI ranks and parallelization

• warpx.numprocs (2 ints for 2D, 3 ints for 3D) optional (default none)

  This optional parameter can be used to control the domain decomposition on the coarsest level. The domain will be chopped into the exact number of pieces in each dimension as specified by this parameter. If it’s not specified, the domain decomposition will be determined by the parameters that will be discussed below. If specified, the product of the numbers must be equal to the number of MPI processes.

• amr.max_grid_size (integer) optional (default 128)

  Maximum allowable size of each subdomain (expressed in number of grid points, in each direction). Each subdomain has its own ghost cells, and can be handled by a different MPI rank ; several OpenMP threads can work simultaneously on the same subdomain.

  If max_grid_size is such that the total number of subdomains is larger that the number of MPI ranks used, than some MPI ranks will handle several subdomains, thereby providing additional flexibility for load balancing.

  When using mesh refinement, this number applies to the subdomains of the coarsest level, but also to any of the finer level.

• algo.load_balance_intervals (string) optional (default 0)

  Using the Intervals parser syntax, this string defines the timesteps at which WarpX should try to redistribute the work across MPI ranks, in order to have better load balancing. Use 0 to disable load_balancing.

  When performing load balancing, WarpX measures the wall time for computational parts of the PIC cycle. It then uses this data to decide how to redistribute the subdomains across MPI ranks. (Each subdomain is unchanged, but its owner is changed in order to have better performance.) This relies on each MPI rank handling several (in fact many) subdomains (see max_grid_size).

• algo.load_balance_efficiency_ratio_threshold (float) optional (default 1.1)

  Controls whether to adopt a proposed distribution mapping computed during a load balance. If the the ratio of the proposed to current distribution mapping efficiency (i.e., average cost per MPI process; efficiency is a number in the range [0, 1]) is greater than the threshold value, the proposed distribution mapping is adopted. The suggested range of values is algo.load_balance_efficiency_ratio_threshold >= 1, which ensures that the new distribution mapping is adopted only if doing so would improve the load balance efficiency. The higher the threshold value, the more conservative is the criterion for adoption of a proposed distribution; for example, with algo.load_balance_efficiency_ratio_threshold = 1, the proposed distribution is adopted any time the proposed distribution improves load balancing; if instead algo.load_balance_efficiency_ratio_threshold = 2, the proposed distribution is adopted only if doing so would yield a 100% to the load balance efficiency (with this threshold value, if the current efficiency is 0.45, the new distribution would only be adopted if the proposed efficiency were greater than 0.9).

• algo.load_balance_with_sfc (0 or 1) optional (default 0)

  If this is 1: use a Space-Filling Curve (SFC) algorithm in order to perform load-balancing of the simulation. If this is 0: the Knapsack algorithm is used instead.

• algo.load_balance_knapsack_factor (float) optional (default 1.24)

  Controls the maximum number of boxes that can be assigned to a rank during load balance when using the ‘knapsack’ policy for update of the distribution mapping; the maximum is load_balance_knapsack_factor*(average number of boxes per rank). For example, if there are 4 boxes per rank and load_balance_knapsack_factor=2, no more than 8 boxes can be assigned to any rank.

• algo.load_balance_costs_update (heuristic or timers) optional (default timers)

  If this is heuristic: load balance costs are updated according to a measure of particles and cells assigned to each box of the domain. The cost \(c\) is computed as

  \[c = n_{\text{particle}} \cdot w_{\text{particle}} + n_{\text{cell}} \cdot w_{\text{cell}},\]

  where \(n_{\text{particle}}\) is the number of particles on the box, \(w_{\text{particle}}\) is the particle cost weight factor (controlled by algo.costs_heuristic_particles_wt), \(n_{\text{cell}}\) is the number of cells on the box, and \(w_{\text{cell}}\) is the cell cost weight factor (controlled by algo.costs_heuristic_cells_wt).

  If this is timers: costs are updated according to in-code timers.

• algo.costs_heuristic_particles_wt (float) optional

  Particle weight factor used in Heuristic strategy for costs update; if running on GPU, the particle weight is set to a value determined from single-GPU tests on Summit, depending on the choice of solver (FDTD or PSATD) and order of the particle shape. If running on CPU, the default value is 0.9. If running on GPU, the default value is

  	Particle shape factor
  	1	2	3
  FDTD/CKC	0.599	0.732	0.855
  PSATD	0.425	0.595	0.75

• algo.costs_heuristic_cells_wt (float) optional

  Cell weight factor used in Heuristic strategy for costs update; if running on GPU, the cell weight is set to a value determined from single-GPU tests on Summit, depending on the choice of solver (FDTD or PSATD) and order of the particle shape. If running on CPU, the default value is 0.1. If running on GPU, the default value is

  	Particle shape factor
  	1	2	3
  FDTD/CKC	0.401	0.268	0.145
  PSATD	0.575	0.405	0.25

• warpx.do_dynamic_scheduling (0 or 1) optional (default 1)

  Whether to activate OpenMP dynamic scheduling.

Math parser and user-defined constants

WarpX uses AMReX’s math parser that reads expressions in the input file. It can be used in all input parameters that consist of one or more integers or floats. Integer input expecting boolean, 0 or 1, are not parsed. Note that when multiple values are expected, the expressions are space delimited. For integer input values, the expressions are evaluated as real numbers and the final result rounded to the nearest integer. See this section of the AMReX documentation for a complete list of functions supported by the math parser.

WarpX constants

WarpX provides a few pre-defined constants, that can be used for any parameter that consists of one or more floats.

q_e	elementary charge
m_e	electron mass
m_p	proton mass
m_u	unified atomic mass unit (Dalton)
epsilon0	vacuum permittivity
mu0	vacuum permeability
clight	speed of light
kb	Boltzmann’s constant (J/K)
pi	math constant pi

See Source/Utils/WarpXConst.H for the values.

User-defined constants

Users can define their own constants in the input file. These constants can be used for any parameter that consists of one or more integers or floats. User-defined constant names can contain only letters, numbers and the character _. The name of each constant has to begin with a letter. The following names are used by WarpX, and cannot be used as user-defined constants: x, y, z, X, Y, t. The values of the constants can include the predefined WarpX constants listed above as well as other user-defined constants. For example:

• my_constants.a0 = 3.0

• my_constants.z_plateau = 150.e-6

• my_constants.n0 = 1.e22

• my_constants.wp = sqrt(n0*q_e**2/(epsilon0*m_e))

Coordinates

Besides, for profiles that depend on spatial coordinates (the plasma momentum distribution or the laser field, see below Particle initialization and Laser initialization), the parser will interpret some variables as spatial coordinates. These are specified in the input parameter, i.e., density_function(x,y,z) and field_function(X,Y,t).

The parser reads python-style expressions between double quotes, for instance "a0*x**2 * (1-y*1.e2) * (x>0)" is a valid expression where a0 is a user-defined constant (see above) and x and y are spatial coordinates. The names are case sensitive. The factor (x>0) is 1 where x>0 and 0 where x<=0. It allows the user to define functions by intervals. Alternatively the expression above can be written as if(x>0, a0*x**2 * (1-y*1.e2), 0).

Particle initialization

• particles.species_names (strings, separated by spaces)

  The name of each species. This is then used in the rest of the input deck ; in this documentation we use <species_name> as a placeholder.

• particles.photon_species (strings, separated by spaces)

  List of species that are photon species, if any. This is required when compiling with QED=TRUE.

• particles.use_fdtd_nci_corr (0 or 1) optional (default 0)

  Whether to activate the FDTD Numerical Cherenkov Instability corrector. Not currently available in the RZ configuration.

• particles.rigid_injected_species (strings, separated by spaces)

  List of species injected using the rigid injection method. The rigid injection method is useful when injecting a relativistic particle beam in boosted-frame simulations; see the input-output section for more details. For species injected using this method, particles are translated along the +z axis with constant velocity as long as their z coordinate verifies z<zinject_plane. When z>zinject_plane, particles are pushed in a standard way, using the specified pusher. (see the parameter <species_name>.zinject_plane below)

• particles.do_tiling (bool) optional (default false if WarpX is compiled for GPUs, true otherwise)

  Controls whether tiling (‘cache blocking’) transformation is used for particles. Tiling should be on when using OpenMP and off when using GPUs.

• <species_name>.species_type (string) optional (default unspecified)

  Type of physical species. Currently, the accepted species are "electron", "positron", "muon", "antimuon", "photon", "neutron", "proton" , "alpha", "hydrogen1" (a.k.a. "protium"), "hydrogen2" (a.k.a. "deuterium"), "hydrogen3" (a.k.a. "tritium"), "helium", "helium3", "helium4", "lithium", "lithium6", "lithium7", "beryllium", "beryllium9", "boron", "boron10", "boron11", "carbon", "carbon12", "carbon13", "carbon14", "nitrogen", "nitrogen14", "nitrogen15", "oxygen", "oxygen16", "oxygen17", "oxygen18", "fluorine", "fluorine19", "neon", "neon20", "neon21", "neon22", "aluminium", "argon", "copper", "xenon" and "gold". The difference between "proton" and "hydrogen1" is that the mass of the latter includes also the mass of the bound electron (same for "alpha" and "helium4"). When only the name of an element is specified, the mass is a weighted average of the masses of the stable isotopes. For all the elements with Z < 11 we provide also the stable isotopes as an option for species_type (e.g., "helium3" and "helium4"). Either species_type or both mass and charge have to be specified.

• <species_name>.charge (float) optional (default NaN)

  The charge of one physical particle of this species. If species_type is specified, the charge will be set to the physical value and charge is optional. When <species>.do_field_ionization = 1, the physical particle charge is equal to ionization_initial_level * charge, so latter parameter should be equal to q_e (which is defined in WarpX as the elementary charge in coulombs).

• <species_name>.mass (float) optional (default NaN)

  The mass of one physical particle of this species. If species_type is specified, the mass will be set to the physical value and mass is optional.

• <species_name>.xmin,ymin,zmin and <species_name>.xmax,ymax,zmax (float) optional (default unlimited)

  When <species_name>.xmin and <species_name>.xmax are set, they delimit the region within which particles are injected. If periodic boundary conditions are used in direction i, then the default (i.e. if the range is not specified) range will be the simulation box, [geometry.prob_hi[i], geometry.prob_lo[i]].

• <species_name>.injection_sources (list of strings) optional

  Names of additional injection sources. By default, WarpX assumes one injection source per species, hence all of the input parameters below describing the injection are parameters directly of the species. However, this option allows additional sources, the names of which are specified here. For each source, the name of the source is added to the input parameters below. For instance, with <species_name>.injection_sources = source1 source2 there can be the two input parameters <species_name>.source1.injection_style and <species_name>.source2.injection_style. For the parameters of each source, the parameter with the name of the source will be used. If it is not given, the value of the parameter without the source name will be used. This allows parameters used for all sources to be specified once. For example, if the source1 and source2 have the same value of uz_m, then it can be set using <species_name>.uz_m instead of setting it for each source. Note that since by default <species_name>.injection_style = none, all injection sources can be input this way. Note that if a moving window is used, the bulk velocity of all of the sources must be the same since it is used when updating the window.

• <species_name>.injection_style (string; default: none)

  Determines how the (macro-)particles will be injected in the simulation. The number of particles per cell is always given with respect to the coarsest level (level 0/mother grid), even if particles are immediately assigned to a refined patch.

  The options are:

  • NUniformPerCell: injection with a fixed number of evenly-spaced particles per cell. This requires the additional parameter <species_name>.num_particles_per_cell_each_dim.

  • NRandomPerCell: injection with a fixed number of randomly-distributed particles per cell. This requires the additional parameter <species_name>.num_particles_per_cell.

  • SingleParticle: Inject a single macroparticle. This requires the additional parameters:

    • <species_name>.single_particle_pos (3 doubles, particle 3D position [meter])

    • <species_name>.single_particle_u (3 doubles, particle 3D normalized momentum, i.e. \(\gamma \beta\))

    • <species_name>.single_particle_weight ( double, macroparticle weight, i.e. number of physical particles it represents)

  • MultipleParticles: Inject multiple macroparticles. This requires the additional parameters:

    • <species_name>.multiple_particles_pos_x (list of doubles, X positions of the particles [meter])

    • <species_name>.multiple_particles_pos_y (list of doubles, Y positions of the particles [meter])

    • <species_name>.multiple_particles_pos_z (list of doubles, Z positions of the particles [meter])

    • <species_name>.multiple_particles_ux (list of doubles, X normalized momenta of the particles, i.e. \(\gamma \beta_x\))

    • <species_name>.multiple_particles_uy (list of doubles, Y normalized momenta of the particles, i.e. \(\gamma \beta_y\))

    • <species_name>.multiple_particles_uz (list of doubles, Z normalized momenta of the particles, i.e. \(\gamma \beta_z\))

    • <species_name>.multiple_particles_weight (list of doubles, macroparticle weights, i.e. number of physical particles each represents)

  • gaussian_beam: Inject particle beam with gaussian distribution in space in all directions. This requires additional parameters:

    • <species_name>.q_tot (beam charge),

    • <species_name>.npart (number of macroparticles in the beam),

    • <species_name>.x/y/z_m (average position in x/y/z),

    • <species_name>.x/y/z_rms (standard deviation in x/y/z),

    There are additional optional parameters:

    • <species_name>.x/y/z_cut (optional, particles with abs(x-x_m) > x_cut*x_rms are not injected, same for y and z. <species_name>.q_tot is the charge of the un-cut beam, so that cutting the distribution is likely to result in a lower total charge),

    • <species_name>.do_symmetrize (optional, whether to symmetrize the beam)

    • <species_name>.symmetrization_order (order of symmetrization, default is 4, can be 4 or 8).

    If <species_name>.do_symmetrize is 0, no symmetrization occurs. If <species_name>.do_symmetrize is 1, then the beam is symmetrized according to the value of <species_name>.symmetrization_order. If set to 4, symmetrization is in the x and y direction, (x,y) (-x,y) (x,-y) (-x,-y). If set to 8, symmetrization is also done with x and y exchanged, (y,x), (-y,x), (y,-x), (-y,-x)).

    • <species_name>.focal_distance (optional, distance between the beam centroid and the position of the focal plane of the beam, along the direction of the beam mean velocity; space charge is ignored in the initialization of the particles)

    If <species_name>.focal_distance is specified, x_rms, y_rms and z_rms are the sizes of the beam in the focal plane. Since the beam is not necessarily initialized close to its focal plane, the initial size of the beam will differ from x_rms, y_rms, z_rms.

    Usually, in accelerator physics the operative quantities are the normalized emittances \(\epsilon_{x,y}\) and beta functions \(\beta_{x,y}\). We assume that the beam travels along \(z\) and we mark the quantities evaluated at the focal plane with a \(*\). Therefore, the normalized transverse emittances and beta functions are related to the focal distance \(f = z - z^*\), the beam sizes \(\sigma_{x,y}\) (which in the code are x_rms, y_rms), the beam relativistic Lorentz factor \(\gamma\), and the normalized momentum spread \(\Delta u_{x,y}\) according to the equations below (Wiedemann [3]).

    \[ \begin{align}\begin{aligned}\Delta u_{x,y} &= \frac{\epsilon^*_{x,y}}{\sigma^*_{x,y}},\\\sigma*_{x, y} &= \sqrt{ \frac{ \epsilon^*_{x,y} \beta^*_{x,y} }{\gamma}},\\\sigma_{x,y}(z) &= \sigma^*_{x,y} \sqrt{1 + \left( \frac{z - z^*}{\beta^*_{x,y}} \right)^2}\end{aligned}\end{align} \]

  • external_file: Inject macroparticles with properties (mass, charge, position, and momentum - \(\gamma \beta m c\)) read from an external openPMD file. With it users can specify the additional arguments:

    • <species_name>.injection_file (string) openPMD file name and

    • <species_name>.charge (double) optional (default is read from openPMD file) when set this will be the charge of the physical particle represented by the injected macroparticles.

    • <species_name>.mass (double) optional (default is read from openPMD file) when set this will be the charge of the physical particle represented by the injected macroparticles.

    • <species_name>.z_shift (double) optional (default is no shift) when set this value will be added to the longitudinal, z, position of the particles.

    • <species_name>.impose_t_lab_from_file (bool) optional (default is false) only read if warpx.gamma_boost > 1., it allows to set t_lab for the Lorentz Transform as being the time stored in the openPMD file.

    Warning: q_tot!=0 is not supported with the external_file injection style. If a value is provided, it is ignored and no re-scaling is done. The external file must include the species openPMD::Record labeled position and momentum (double arrays), with dimensionality and units set via openPMD::setUnitDimension and setUnitSI. If the external file also contains openPMD::Records for mass and charge (constant double scalars) then the species will use these, unless overwritten in the input file (see <species_name>.mass, <species_name>.charge or <species_name>.species_type). The external_file option is currently implemented for 2D, 3D and RZ geometries, with record components in the cartesian coordinates (x,y,z) for 3D and RZ, and (x,z) for 2D. For more information on the openPMD format and how to build WarpX with it, please visit the install section.

  • NFluxPerCell: Continuously inject a flux of macroparticles from a planar surface. This requires the additional parameters:

    • <species_name>.flux_profile (see the description of this parameter further below)

    • <species_name>.surface_flux_pos (double, location of the injection plane [meter])

    • <species_name>.flux_normal_axis (x, y, or z for 3D, x or z for 2D, or r, t, or z for RZ. When flux_normal_axis is r or t, the x and y components of the user-specified momentum distribution are interpreted as the r and t components respectively)

    • <species_name>.flux_direction (-1 or +1, direction of flux relative to the plane)

    • <species_name>.num_particles_per_cell (double)

    • <species_name>.flux_tmin (double, Optional time at which the flux will be turned on. Ignored when negative.)

    • <species_name>.flux_tmax (double, Optional time at which the flux will be turned off. Ignored when negative.)

  • none: Do not inject macro-particles (for example, in a simulation that starts with neutral, ionizable atoms, one may want to create the electrons species – where ionized electrons can be stored later on – without injecting electron macro-particles).

• <species_name>.num_particles_per_cell_each_dim (3 integers in 3D and RZ, 2 integers in 2D)

  With the NUniformPerCell injection style, this specifies the number of particles along each axis within a cell. Note that for RZ, the three axis are radius, theta, and z and that the recommended number of particles per theta is at least two times the number of azimuthal modes requested. (It is recommended to do a convergence scan of the number of particles per theta)

• <species_name>.random_theta (bool) optional (default 1)

  When using RZ geometry, whether to randomize the azimuthal position of particles. This is used when <species_name>.injection_style = NUniformPerCell.

• <species_name>.do_splitting (bool) optional (default 0)

  Split particles of the species when crossing the boundary from a lower resolution domain to a higher resolution domain.

  Currently implemented on CPU only.

• <species_name>.do_continuous_injection (0 or 1)

  Whether to inject particles during the simulation, and not only at initialization. This can be required with a moving window and/or when running in a boosted frame.

• <species_name>.initialize_self_fields (0 or 1)

  Whether to calculate the space-charge fields associated with this species at the beginning of the simulation. The fields are calculated for the mean gamma of the species.

• <species_name>.self_fields_required_precision (float, default: 1.e-11)

  The relative precision with which the initial space-charge fields should be calculated. More specifically, the initial space-charge fields are computed with an iterative Multi-Level Multi-Grid (MLMG) solver. For highly-relativistic beams, this solver can fail to reach the default precision within a reasonable time ; in that case, users can set a relaxed precision requirement through self_fields_required_precision.

• <species_name>.self_fields_absolute_tolerance (float, default: 0.0)

  The absolute tolerance with which the space-charge fields should be calculated in units of \(\mathrm{V/m}^2\). More specifically, the acceptable residual with which the solution can be considered converged. In general this should be left as the default, but in cases where the simulation state changes very little between steps it can occur that the initial guess for the MLMG solver is so close to the converged value that it fails to improve that solution sufficiently to reach the self_fields_required_precision value.

• <species_name>.self_fields_max_iters (integer, default: 200)

  Maximum number of iterations used for MLMG solver for initial space-charge fields calculation. In case if MLMG converges but fails to reach the desired self_fields_required_precision, this parameter may be increased.

• <species_name>.profile (string)

  Density profile for this species. The options are:

  • constant: Constant density profile within the box, or between <species_name>.xmin and <species_name>.xmax (and same in all directions). This requires additional parameter <species_name>.density. i.e., the plasma density in \(m^{-3}\).

  • predefined: Predefined density profile. This requires additional parameters <species_name>.predefined_profile_name and <species_name>.predefined_profile_params. Currently, only a parabolic channel density profile is implemented.

  • parse_density_function: the density is given by a function in the input file. It requires additional argument <species_name>.density_function(x,y,z), which is a mathematical expression for the density of the species, e.g. electrons.density_function(x,y,z) = "n0+n0*x**2*1.e12" where n0 is a user-defined constant, see above. WARNING: where density_function(x,y,z) is close to zero, particles will still be injected between xmin and xmax etc., with a null weight. This is undesirable because it results in useless computing. To avoid this, see option density_min below.

• <species_name>.flux_profile (string)

  Defines the expression of the flux, when using <species_name>.injection_style=NFluxPerCell

  • constant: Constant flux. This requires the additional parameter <species_name>.flux. i.e., the injection flux in \(m^{-2}.s^{-1}\).

  • parse_flux_function: the flux is given by a function in the input file. It requires the additional argument <species_name>.flux_function(x,y,z,t), which is a mathematical expression for the flux of the species.

• <species_name>.density_min (float) optional (default 0.)

  Minimum plasma density. No particle is injected where the density is below this value.

• <species_name>.density_max (float) optional (default infinity)

  Maximum plasma density. The density at each point is the minimum between the value given in the profile, and density_max.

• <species_name>.radially_weighted (bool) optional (default true)

  Whether particle’s weight is varied with their radius. This only applies to cylindrical geometry. The only valid value is true.

• <species_name>.momentum_distribution_type (string)

  Distribution of the normalized momentum (u=p/mc) for this species. The options are:

  • at_rest: Particles are initialized with zero momentum.

  • constant: constant momentum profile. This can be controlled with the additional parameters <species_name>.ux, <species_name>.uy and <species_name>.uz, the normalized momenta in the x, y and z direction respectively, which are all 0. by default.

  • uniform: uniform probability distribution between a minimum and a maximum value. The x, y and z directions are sampled independently and the final momentum space is a cuboid. The parameters that control the minimum and maximum domain of the distribution are <species_name>.u<x,y,z>_min and <species_name>.u<x,y,z>_max in each direction respectively (e.g., <species_name>.uz_min = 0.2 and <species_name>.uz_max = 0.4 to control the generation along the z direction). All the parameters default to 0.

  • gaussian: gaussian momentum distribution in all 3 directions. This can be controlled with the additional arguments for the average momenta along each direction <species_name>.ux_m, <species_name>.uy_m and <species_name>.uz_m as well as standard deviations along each direction <species_name>.ux_th, <species_name>.uy_th and <species_name>.uz_th. These 6 parameters are all 0. by default.

  • gaussianflux: Gaussian momentum flux distribution, which is Gaussian in the plane and v*Gaussian normal to the plane. It can only be used when injection_style = NFluxPerCell. This can be controlled with the additional arguments to specify the plane’s orientation, <species_name>.flux_normal_axis and <species_name>.flux_direction, for the average momenta along each direction <species_name>.ux_m, <species_name>.uy_m and <species_name>.uz_m, as well as standard deviations along each direction <species_name>.ux_th, <species_name>.uy_th and <species_name>.uz_th. ux_m, uy_m, uz_m, ux_th, uy_th and uz_th are all 0. by default.

  • maxwell_boltzmann: Maxwell-Boltzmann distribution that takes a dimensionless temperature parameter \(\theta\) as an input, where \(\theta = \frac{k_\mathrm{B} \cdot T}{m \cdot c^2}\), \(T\) is the temperature in Kelvin, \(k_\mathrm{B}\) is the Boltzmann constant, \(c\) is the speed of light, and \(m\) is the mass of the species. Theta is specified by a combination of <species_name>.theta_distribution_type, <species_name>.theta, and <species_name>.theta_function(x,y,z) (see below). For values of \(\theta > 0.01\), errors due to ignored relativistic terms exceed 1%. Temperatures less than zero are not allowed. The plasma can be initialized to move at a bulk velocity \(\beta = v/c\). The speed is specified by the parameters <species_name>.beta_distribution_type, <species_name>.beta, and <species_name>.beta_function(x,y,z) (see below). \(\beta\) can be positive or negative and is limited to the range \(-1 < \beta < 1\). The direction of the velocity field is given by <species_name>.bulk_vel_dir = (+/-) 'x', 'y', 'z', and must be the same across the domain. Please leave no whitespace between the sign and the character on input. A direction without a sign will be treated as positive. The MB distribution is initialized in the drifting frame by sampling three Gaussian distributions in each dimension using, the Box Mueller method, and then the distribution is transformed to the simulation frame using the flipping method. The flipping method can be found in Zenitani 2015 section III. B. (Phys. Plasmas 22, 042116). By default, beta is equal to 0. and bulk_vel_dir is +x.

    Note that though the particles may move at relativistic speeds in the simulation frame, they are not relativistic in the drift frame. This is as opposed to the Maxwell Juttner setting, which initializes particles with relativistic momentums in their drifting frame.

  • maxwell_juttner: Maxwell-Juttner distribution for high temperature plasma that takes a dimensionless temperature parameter \(\theta\) as an input, where \(\theta = \frac{k_\mathrm{B} \cdot T}{m \cdot c^2}\), \(T\) is the temperature in Kelvin, \(k_\mathrm{B}\) is the Boltzmann constant, and \(m\) is the mass of the species. Theta is specified by a combination of <species_name>.theta_distribution_type, <species_name>.theta, and <species_name>.theta_function(x,y,z) (see below). The Sobol method used to generate the distribution will not terminate for \(\theta \lesssim 0.1\), and the code will abort if it encounters a temperature below that threshold. The Maxwell-Boltzmann distribution is recommended for temperatures in the range \(0.01 < \theta < 0.1\). Errors due to relativistic effects can be expected to approximately between 1% and 10%. The plasma can be initialized to move at a bulk velocity \(\beta = v/c\). The speed is specified by the parameters <species_name>.beta_distribution_type, <species_name>.beta, and <species_name>.beta_function(x,y,z) (see below). \(\beta\) can be positive or negative and is limited to the range \(-1 < \beta < 1\). The direction of the velocity field is given by <species_name>.bulk_vel_dir = (+/-) 'x', 'y', 'z', and must be the same across the domain. Please leave no whitespace between the sign and the character on input. A direction without a sign will be treated as positive. The MJ distribution will be initialized in the moving frame using the Sobol method, and then the distribution will be transformed to the simulation frame using the flipping method. Both the Sobol and the flipping method can be found in Zenitani 2015 (Phys. Plasmas 22, 042116). By default, beta is equal to 0. and bulk_vel_dir is +x.

    Please take notice that particles initialized with this setting can be relativistic in two ways. In the simulation frame, they can drift with a relativistic speed beta. Then, in the drifting frame they are still moving with relativistic speeds due to high temperature. This is as opposed to the Maxwell Boltzmann setting, which initializes non-relativistic plasma in their relativistic drifting frame.

  • radial_expansion: momentum depends on the radial coordinate linearly. This can be controlled with additional parameter u_over_r which is the slope (0. by default).

  • parse_momentum_function: the momentum \(u = (u_{x},u_{y},u_{z})=(\gamma v_{x}/c,\gamma v_{y}/c,\gamma v_{z}/c)\) is given by a function in the input file. It requires additional arguments <species_name>.momentum_function_ux(x,y,z), <species_name>.momentum_function_uy(x,y,z) and <species_name>.momentum_function_uz(x,y,z), which gives the distribution of each component of the momentum as a function of space.

  • gaussian_parse_momentum_function: Gaussian momentum distribution where the mean and the standard deviation are given by functions of position in the input file. Both are assumed to be non-relativistic. The mean is the normalized momentum, \(u_m = \gamma v_m/c\). The standard deviation is normalized, \(u_th = v_th/c\). For example, this might be u_th = sqrt(T*q_e/mass)/clight given the temperature (in eV) and mass. It requires the following arguments:

    • <species_name>.momentum_function_ux_m(x,y,z): mean \(u_{x}\)

    • <species_name>.momentum_function_uy_m(x,y,z): mean \(u_{y}\)

    • <species_name>.momentum_function_uz_m(x,y,z): mean \(u_{z}\)

    • <species_name>.momentum_function_ux_th(x,y,z): standard deviation of \(u_{x}\)

    • <species_name>.momentum_function_uy_th(x,y,z): standard deviation of \(u_{y}\)

    • <species_name>.momentum_function_uz_th(x,y,z): standard deviation of \(u_{z}\)

• <species_name>.theta_distribution_type (string) optional (default constant)

  Only read if <species_name>.momentum_distribution_type is maxwell_boltzmann or maxwell_juttner. See documentation for these distributions (above) for constraints on values of theta. Temperatures less than zero are not allowed.

  • If constant, use a constant temperature, given by the required float parameter <species_name>.theta.

  • If parser, use a spatially-dependent analytic parser function, given by the required parameter <species_name>.theta_function(x,y,z).

• <species_name>.beta_distribution_type (string) optional (default constant)

  Only read if <species_name>.momentum_distribution_type is maxwell_boltzmann or maxwell_juttner. See documentation for these distributions (above) for constraints on values of beta.

  • If constant, use a constant speed, given by the required float parameter <species_name>.beta.

  • If parser, use a spatially-dependent analytic parser function, given by the required parameter <species_name>.beta_function(x,y,z).

• <species_name>.zinject_plane (float)

  Only read if <species_name> is in particles.rigid_injected_species. Injection plane when using the rigid injection method. See particles.rigid_injected_species above.

• <species_name>.rigid_advance (bool)

  Only read if <species_name> is in particles.rigid_injected_species.

  • If false, each particle is advanced with its own velocity vz until it reaches zinject_plane.

  • If true, each particle is advanced with the average speed of the species vzbar until it reaches zinject_plane.

• species_name.predefined_profile_name (string)

  Only read if <species_name>.profile is predefined.

  • If parabolic_channel, the plasma profile is a parabolic profile with cosine-like ramps at the beginning and the end of the profile. The density is given by

    \[n = n_0 n(x,y) n(z-z_0)\]

    with

    \[n(x,y) = 1 + 4\frac{x^2+y^2}{k_p^2 R_c^4}\]

    where \(k_p\) is the plasma wavenumber associated with density \(n_0\). Here, with \(z_0\) as the start of the plasma, \(n(z-z_0)\) is a cosine-like up-ramp from \(0\) to \(L_{ramp,up}\), constant to \(1\) from \(L_{ramp,up}\) to \(L_{ramp,up} + L_{plateau}\) and a cosine-like down-ramp from \(L_{ramp,up} + L_{plateau}\) to \(L_{ramp,up} + L_{plateau}+L_{ramp,down}\). All parameters are given in predefined_profile_params.

• <species_name>.predefined_profile_params (list of float)

  Parameters for the predefined profiles.

  • If species_name.predefined_profile_name is parabolic_channel, predefined_profile_params contains a space-separated list of the following parameters, in this order: \(z_0\) \(L_{ramp,up}\) \(L_{plateau}\) \(L_{ramp,down}\) \(R_c\) \(n_0\)

• <species_name>.do_backward_propagation (bool)

  Inject a backward-propagating beam to reduce the effect of charge-separation fields when running in the boosted frame. See examples.

• <species_name>.split_type (int) optional (default 0)

  Splitting technique. When 0, particles are split along the simulation axes (4 particles in 2D, 6 particles in 3D). When 1, particles are split along the diagonals (4 particles in 2D, 8 particles in 3D).

• <species_name>.do_not_deposit (0 or 1 optional; default 0)

  If 1 is given, both charge deposition and current deposition will not be done, thus that species does not contribute to the fields.

• <species_name>.do_not_gather (0 or 1 optional; default 0)

  If 1 is given, field gather from grids will not be done, thus that species will not be affected by the field on grids.

• <species_name>.do_not_push (0 or 1 optional; default 0)

  If 1 is given, this species will not be pushed by any pusher during the simulation.

• <species_name>.addIntegerAttributes (list of string)

  User-defined integer particle attribute for species, species_name. These integer attributes will be initialized with user-defined functions when the particles are generated. If the user-defined integer attribute is <int_attrib_name> then the following required parameter must be specified to initialize the attribute. * <species_name>.attribute.<int_attrib_name>(x,y,z,ux,uy,uz,t) (string) t represents the physical time in seconds during the simulation. x, y, z represent particle positions in the unit of meter. ux, uy, uz represent the particle momenta in the unit of \(\gamma v/c\), where \(\gamma\) is the Lorentz factor, \(v/c\) is the particle velocity normalized by the speed of light. E.g. If electrons.addIntegerAttributes = upstream and electrons.upstream(x,y,z,ux,uy,uz,t) = (x>0.0)*1 is provided then, an integer attribute upstream is added to all electron particles and when these particles are generated, the particles with position less than 0 are assigned a value of 1.

• <species_name>.addRealAttributes (list of string)

  User-defined real particle attribute for species, species_name. These real attributes will be initialized with user-defined functions when the particles are generated. If the user-defined real attribute is <real_attrib_name> then the following required parameter must be specified to initialize the attribute.

  • <species_name>.attribute.<real_attrib_name>(x,y,z,ux,uy,uz,t) (string) t represents the physical time in seconds during the simulation. x, y, z represent particle positions in the unit of meter. ux, uy, uz represent the particle momenta in the unit of \(\gamma v/c\), where \(\gamma\) is the Lorentz factor, \(v/c\) is the particle velocity normalized by the speed of light.

• <species>.save_particles_at_xlo/ylo/zlo, <species>.save_particles_at_xhi/yhi/zhi and <species>.save_particles_at_eb (0 or 1 optional, default 0)

  If 1 particles of this species will be copied to the scraped particle buffer for the specified boundary if they leave the simulation domain in the specified direction. If USE_EB=TRUE the save_particles_at_eb flag can be set to 1 to also save particle data for the particles of this species that impact the embedded boundary. The scraped particle buffer can be used to track particle fluxes out of the simulation. The particle data can be written out by setting up a BoundaryScrapingDiagnostic. It is also accessible via the Python interface. The function get_particle_boundary_buffer, found in the picmi.Simulation class as sim.extension.get_particle_boundary_buffer(), can be used to access the scraped particle buffer. An entry is included for every particle in the buffer of the timestep at which the particle was scraped. This can be accessed by passing the argument comp_name="step_scraped" to the above mentioned function.

  Note

  When accessing the data via Python, the scraped particle buffer relies on the user to clear the buffer after processing the data. The buffer will grow unbounded as particles are scraped and therefore could lead to memory issues if not periodically cleared. To clear the buffer call clear_buffer().

• <species>.do_field_ionization (0 or 1) optional (default 0)

  Do field ionization for this species (using the ADK theory).

• <species>.do_adk_correction (0 or 1) optional (default 0)

  Whether to apply the correction to the ADK theory proposed by Zhang, Lan and Lu in Q. Zhang et al. (Phys. Rev. A 90, 043410, 2014). If so, the probability of ionization is modified using an empirical model that should be more accurate in the regime of high electric fields. Currently, this is only implemented for Hydrogen, although Argon is also available in the same reference.

• <species>.physical_element (string)

  Only read if do_field_ionization = 1. Symbol of chemical element for this species. Example: for Helium, use physical_element = He. All the elements up to atomic number Z=100 (Fermium) are supported.

• <species>.ionization_product_species (string)

  Only read if do_field_ionization = 1. Name of species in which ionized electrons are stored. This species must be created as a regular species in the input file (in particular, it must be in particles.species_names).

• <species>.ionization_initial_level (int) optional (default 0)

  Only read if do_field_ionization = 1. Initial ionization level of the species (must be smaller than the atomic number of chemical element given in physical_element).

• <species>.do_classical_radiation_reaction (int) optional (default 0)

  Enables Radiation Reaction (or Radiation Friction) for the species. Species must be either electrons or positrons. Boris pusher must be used for the simulation. If both <species>.do_classical_radiation_reaction and <species>.do_qed_quantum_sync are enabled, then the classical module will be used when the particle’s chi parameter is below qed_qs.chi_min, the discrete quantum module otherwise.

• <species>.do_qed_quantum_sync (int) optional (default 0)

  Enables Quantum synchrotron emission for this species. Quantum synchrotron lookup table should be either generated or loaded from disk to enable this process (see “Lookup tables for QED modules” section below). <species> must be either an electron or a positron species. This feature requires to compile with QED=TRUE

• <species>.do_qed_breit_wheeler (int) optional (default 0)

  Enables non-linear Breit-Wheeler process for this species. Breit-Wheeler lookup table should be either generated or loaded from disk to enable this process (see “Lookup tables for QED modules” section below). <species> must be a photon species. This feature requires to compile with QED=TRUE

• <species>.qed_quantum_sync_phot_product_species (string)

  If an electron or a positron species has the Quantum synchrotron process, a photon product species must be specified (the name of an existing photon species must be provided) This feature requires to compile with QED=TRUE

• <species>.qed_breit_wheeler_ele_product_species (string)

  If a photon species has the Breit-Wheeler process, an electron product species must be specified (the name of an existing electron species must be provided) This feature requires to compile with QED=TRUE

• <species>.qed_breit_wheeler_pos_product_species (string)

  If a photon species has the Breit-Wheeler process, a positron product species must be specified (the name of an existing positron species must be provided). This feature requires to compile with QED=TRUE

• <species>.do_resampling (0 or 1) optional (default 0)

  If 1 resampling is performed for this species. This means that the number of macroparticles will be reduced at specific timesteps while preserving the distribution function as much as possible (in particular the weight of the remaining particles will be increased on average). This can be useful in situations with continuous creation of particles (e.g. with ionization or with QED effects). At least one resampling trigger (see below) must be specified to actually perform resampling.

• <species>.resampling_algorithm (string) optional (default leveling_thinning)

  The algorithm used for resampling. Currently there is only one option, which is already set by default:

  • leveling_thinning This algorithm is defined in Muraviev et al. [4]. It has two parameters:

    • <species>.resampling_algorithm_target_ratio (float) optional (default 1.5)

      This roughly corresponds to the ratio between the number of particles before and after resampling.

    • <species>.resampling_algorithm_min_ppc (int) optional (default 1)

      Resampling is not performed in cells with a number of macroparticles strictly smaller than this parameter.

• <species>.resampling_trigger_intervals (string) optional (default 0)

  Using the Intervals parser syntax, this string defines timesteps at which resampling is performed.

• <species>.resampling_trigger_max_avg_ppc (float) optional (default infinity)

  Resampling is performed everytime the number of macroparticles per cell of the species averaged over the whole simulation domain exceeds this parameter.

Cold Relativistic Fluid initialization

• fluids.species_names (strings, separated by spaces)

  Defines the names of each fluid species. It is a required input to create and evolve fluid species using the cold relativistic fluid equations. Most of the parameters described in the section “Particle initialization” can also be used to initialize fluid properties (e.g. initial density distribution). For fluid-specific inputs we use <fluid_species_name> as a placeholder. Also see external fields for how to specify these for fluids as the function names differ.

Laser initialization

• lasers.names (list of string)

  Name of each laser. This is then used in the rest of the input deck ; in this documentation we use <laser_name> as a placeholder. The parameters below must be provided for each laser pulse.

• <laser_name>.position (3 floats in 3D and 2D ; in meters)

  The coordinates of one of the point of the antenna that will emit the laser. The plane of the antenna is entirely defined by <laser_name>.position and <laser_name>.direction.

  <laser_name>.position also corresponds to the origin of the coordinates system for the laser tranverse profile. For instance, for a Gaussian laser profile, the peak of intensity will be at the position given by <laser_name>.position. This variable can thus be used to shift the position of the laser pulse transversally.

  Note

  In 2D, <laser_name>.position is still given by 3 numbers, but the second number is ignored.

  When running a boosted-frame simulation, provide the value of <laser_name>.position in the laboratory frame, and use warpx.gamma_boost to automatically perform the conversion to the boosted frame. Note that, in this case, the laser antenna will be moving, in the boosted frame.

• <laser_name>.polarization (3 floats in 3D and 2D)

  The coordinates of a vector that points in the direction of polarization of the laser. The norm of this vector is unimportant, only its direction matters.

  Note

  Even in 2D, all the 3 components of this vectors are important (i.e. the polarization can be orthogonal to the plane of the simulation).

• <laser_name>.direction (3 floats in 3D)

  The coordinates of a vector that points in the propagation direction of the laser. The norm of this vector is unimportant, only its direction matters.

  The plane of the antenna that will emit the laser is orthogonal to this vector.

  Warning

  When running boosted-frame simulations, <laser_name>.direction should be parallel to warpx.boost_direction, for now.

• <laser_name>.e_max (float ; in V/m)

  Peak amplitude of the laser field, in the focal plane.

  For a laser with a wavelength \(\lambda = 0.8\,\mu m\), the peak amplitude is related to \(a_0\) by:

  \[E_{max} = a_0 \frac{2 \pi m_e c^2}{e\lambda} = a_0 \times (4.0 \cdot 10^{12} \;V.m^{-1})\]

  When running a boosted-frame simulation, provide the value of <laser_name>.e_max in the laboratory frame, and use warpx.gamma_boost to automatically perform the conversion to the boosted frame.

• <laser_name>.a0 (float ; dimensionless)

  Peak normalized amplitude of the laser field, in the focal plane (given in the lab frame, just as e_max above). See the description of <laser_name>.e_max for the conversion between a0 and e_max. Either a0 or e_max must be specified.

• <laser_name>.wavelength (float; in meters)

  The wavelength of the laser in vacuum.

  When running a boosted-frame simulation, provide the value of <laser_name>.wavelength in the laboratory frame, and use warpx.gamma_boost to automatically perform the conversion to the boosted frame.

• <laser_name>.profile (string)

  The spatio-temporal shape of the laser. The options that are currently implemented are:

  • "Gaussian": The transverse and longitudinal profiles are Gaussian.

  • "parse_field_function": the laser electric field is given by a function in the input file. It requires additional argument <laser_name>.field_function(X,Y,t), which is a mathematical expression , e.g. <laser_name>.field_function(X,Y,t) = "a0*X**2 * (X>0) * cos(omega0*t)" where a0 and omega0 are a user-defined constant, see above. The profile passed here is the full profile, not only the laser envelope. t is time and X and Y are coordinates orthogonal to <laser_name>.direction (not necessarily the x and y coordinates of the simulation). All parameters above are required, but none of the parameters below are used when <laser_name>.parse_field_function=1. Even though <laser_name>.wavelength and <laser_name>.e_max should be included in the laser function, they still have to be specified as they are used for numerical purposes.

  • "from_file": the electric field of the laser is read from an external file. Currently both the lasy format as well as a custom binary format are supported. It requires to provide the name of the file to load setting the additional parameter <laser_name>.binary_file_name or <laser_name>.lasy_file_name (string). It accepts an optional parameter <laser_name>.time_chunk_size (int), supported for both lasy and binary files; this allows to read only time_chunk_size timesteps from the file. New timesteps are read as soon as they are needed.

    The default value is automatically set to the number of timesteps contained in the file (i.e. only one read is performed at the beginning of the simulation). It also accepts the optional parameter <laser_name>.delay (float; in seconds), which allows delaying (delay > 0) or anticipating (delay < 0) the laser by the specified amount of time.

    Details about the usage of the lasy format: lasy can produce either 3D Cartesian files or RZ files. WarpX can read both types of files independently of the geometry in which it was compiled (e.g. WarpX compiled with WarpX_DIMS=RZ can read 3D Cartesian lasy files). In the case where WarpX is compiled in 2D (or 1D) Cartesian, the laser antenna will emit the field values that correspond to the slice y=0 in the lasy file (and x=0 in the 1D case). One can generate a lasy file from Python, see an example at Examples/Tests/laser_injection_from_file.

    Details about the usage of the binary format: The external binary file should provide E(x,y,t) on a rectangular (necessarily uniform) grid. The code performs a bi-linear (in 2D) or tri-linear (in 3D) interpolation to set the field values. x,y,t are meant to be in S.I. units, while the field value is meant to be multiplied by <laser_name>.e_max (i.e. in most cases the maximum of abs(E(x,y,t)) should be 1, so that the maximum field intensity can be set straightforwardly with <laser_name>.e_max). The binary file has to respect the following format:

    • flag to indicate the grid is uniform (1 byte, 0 means non-uniform, !=0 means uniform) - only uniform is supported

    • nt, number of timesteps (uint32_t, must be >=2)

    • nx, number of points along x (uint32_t, must be >=2)

    • ny, number of points along y (uint32_t, must be 1 for 2D simulations and >=2 for 3D simulations)

    • timesteps (double[2]=[t_min,t_max])

    • x_coords (double[2]=[x_min,x_max])

    • y_coords (double[1] in 2D, double[2]=[y_min,y_max] in 3D)

    • field_data (double[nt x nx * ny], with nt being the slowest coordinate).

    A binary file can be generated from Python, see an example at Examples/Tests/laser_injection_from_file

• <laser_name>.profile_t_peak (float; in seconds)

  The time at which the laser reaches its peak intensity, at the position given by <laser_name>.position (only used for the "gaussian" profile)

  When running a boosted-frame simulation, provide the value of <laser_name>.profile_t_peak in the laboratory frame, and use warpx.gamma_boost to automatically perform the conversion to the boosted frame.

• <laser_name>.profile_duration (float ; in seconds)

  The duration of the laser pulse for the "gaussian" profile, defined as \(\tau\) below:

  \[E(\boldsymbol{x},t) \propto \exp\left( -\frac{(t-t_{peak})^2}{\tau^2} \right)\]

  Note that \(\tau\) relates to the full width at half maximum (FWHM) of intensity, which is closer to pulse length measurements in experiments, as \(\tau = \mathrm{FWHM}_I / \sqrt{2\ln(2)}\) \(\approx \mathrm{FWHM}_I / 1.1774\).

  For a chirped laser pulse (i.e. with a non-zero <laser_name>.phi2), profile_duration is the Fourier-limited duration of the pulse, not the actual duration of the pulse. See the documentation for <laser_name>.phi2 for more detail.

  When running a boosted-frame simulation, provide the value of <laser_name>.profile_duration in the laboratory frame, and use warpx.gamma_boost to automatically perform the conversion to the boosted frame.

• <laser_name>.profile_waist (float ; in meters)

  The waist of the transverse Gaussian \(w_0\), i.e. defined such that the electric field of the laser pulse in the focal plane is of the form:

  \[E(\boldsymbol{x},t) \propto \exp\left( -\frac{\boldsymbol{x}_\perp^2}{w_0^2} \right)\]

• <laser_name>.profile_focal_distance (float; in meters)

  The distance from laser_position to the focal plane. (where the distance is defined along the direction given by <laser_name>.direction.)

  Use a negative number for a defocussing laser instead of a focussing laser.

  When running a boosted-frame simulation, provide the value of <laser_name>.profile_focal_distance in the laboratory frame, and use warpx.gamma_boost to automatically perform the conversion to the boosted frame.

• <laser_name>.phi0 (float; in radians) optional (default 0.)

  The Carrier Envelope Phase, i.e. the phase of the laser oscillation, at the position where the laser envelope is maximum (only used for the "gaussian" profile)

• <laser_name>.stc_direction (3 floats) optional (default 1. 0. 0.)

  Direction of laser spatio-temporal couplings. See definition in Akturk et al. [5].

• <laser_name>.zeta (float; in meters.seconds) optional (default 0.)

  Spatial chirp at focus in direction <laser_name>.stc_direction. See definition in Akturk et al. [5].

• <laser_name>.beta (float; in seconds) optional (default 0.)

  Angular dispersion (or angular chirp) at focus in direction <laser_name>.stc_direction. See definition in Akturk et al. [5].

• <laser_name>.phi2 (float; in seconds**2) optional (default 0.)

  The amount of temporal chirp \(\phi^{(2)}\) at focus (in the lab frame). Namely, a wave packet centered on the frequency \((\omega_0 + \delta \omega)\) will reach its peak intensity at \(z(\delta \omega) = z_0 - c \phi^{(2)} \, \delta \omega\). Thus, a positive \(\phi^{(2)}\) corresponds to positive chirp, i.e. red part of the spectrum in the front of the pulse and blue part of the spectrum in the back. More specifically, the electric field in the focal plane is of the form:

  \[E(\boldsymbol{x},t) \propto Re\left[ \exp\left( -\frac{(t-t_{peak})^2}{\tau^2 + 2i\phi^{(2)}} + i\omega_0 (t-t_{peak}) + i\phi_0 \right) \right]\]

  where \(\tau\) is given by <laser_name>.profile_duration and represents the Fourier-limited duration of the laser pulse. Thus, the actual duration of the chirped laser pulse is:

  \[\tau' = \sqrt{ \tau^2 + 4 (\phi^{(2)})^2/\tau^2 }\]

  See also the definition in Akturk et al. [5].

• <laser_name>.do_continuous_injection (0 or 1) optional (default 0).

  Whether or not to use continuous injection. If the antenna starts outside of the simulation domain but enters it at some point (due to moving window or moving antenna in the boosted frame), use this so that the laser antenna is injected when it reaches the box boundary. If running in a boosted frame, this requires the boost direction, moving window direction and laser propagation direction to be along z. If not running in a boosted frame, this requires the moving window and laser propagation directions to be the same (x, y or z)

• <laser_name>.min_particles_per_mode (int) optional (default 4)

  When using the RZ version, this specifies the minimum number of particles per angular mode. The laser particles are loaded into radial spokes, with the number of spokes given by min_particles_per_mode*(warpx.n_rz_azimuthal_modes-1).

• lasers.deposit_on_main_grid (int) optional (default 0)

  When using mesh refinement, whether the antenna that emits the laser deposits charge/current only on the main grid (i.e. level 0), or also on the higher mesh-refinement levels.

• warpx.num_mirrors (int) optional (default 0)

  Users can input perfect mirror condition inside the simulation domain. The number of mirrors is given by warpx.num_mirrors. The mirrors are orthogonal to the z direction. The following parameters are required when warpx.num_mirrors is >0.

• warpx.mirror_z (list of float) required if warpx.num_mirrors>0

  z location of the front of the mirrors.

• warpx.mirror_z_width (list of float) required if warpx.num_mirrors>0

  z width of the mirrors.

• warpx.mirror_z_npoints (list of int) required if warpx.num_mirrors>0

  In the boosted frame, depending on gamma_boost, warpx.mirror_z_width can be smaller than the cell size, so that the mirror would not work. This parameter is the minimum number of points for the mirror. If mirror_z_width < dz/cell_size, the upper bound of the mirror is increased so that it contains at least mirror_z_npoints.

External fields

Applied to the grid

The external fields defined with input parameters that start with warpx.B_ext_grid_init_ or warpx.E_ext_grid_init_ are applied to the grid directly. In particular, these fields can be seen in the diagnostics that output the fields on the grid.

• When using an electromagnetic field solver, these fields are applied to the grid at the beginning of the simulation, and serve as initial condition for the Maxwell solver.

• When using an electrostatic or magnetostatic field solver, these fields are added to the fields computed by the Poisson solver, at each timestep.

• warpx.B_ext_grid_init_style (string) optional

  This parameter determines the type of initialization for the external magnetic field. By default, the external magnetic field (Bx,By,Bz) is initialized to (0.0, 0.0, 0.0). The string can be set to “constant” if a constant magnetic field is required to be set at initialization. If set to “constant”, then an additional parameter, namely, warpx.B_external_grid must be specified. If set to parse_B_ext_grid_function, then a mathematical expression can be used to initialize the external magnetic field on the grid. It requires additional parameters in the input file, namely, warpx.Bx_external_grid_function(x,y,z), warpx.By_external_grid_function(x,y,z), warpx.Bz_external_grid_function(x,y,z) to initialize the external magnetic field for each of the three components on the grid. Constants required in the expression can be set using my_constants. For example, if warpx.Bx_external_grid_function(x,y,z)=Bo*x + delta*(y + z) then the constants Bo and delta required in the above equation can be set using my_constants.Bo= and my_constants.delta= in the input file. For a two-dimensional simulation, it is assumed that the first dimension is x and the second dimension is z, and the value of y is set to zero. Note that the current implementation of the parser for external B-field does not work with RZ and the code will abort with an error message.

  If B_ext_grid_init_style is set to be read_from_file, an additional parameter, indicating the path of an openPMD data file, warpx.read_fields_from_path must be specified, from which external B field data can be loaded into WarpX. One can refer to input files in Examples/Tests/LoadExternalField for more information. Regarding how to prepare the openPMD data file, one can refer to the openPMD-example-datasets.

• warpx.E_ext_grid_init_style (string) optional

  This parameter determines the type of initialization for the external electric field. By default, the external electric field (Ex,Ey,Ez) to (0.0, 0.0, 0.0). The string can be set to “constant” if a constant electric field is required to be set at initialization. If set to “constant”, then an additional parameter, namely, warpx.E_external_grid must be specified in the input file. If set to parse_E_ext_grid_function, then a mathematical expression can be used to initialize the external electric field on the grid. It required additional parameters in the input file, namely, warpx.Ex_external_grid_function(x,y,z), warpx.Ey_external_grid_function(x,y,z), warpx.Ez_external_grid_function(x,y,z) to initialize the external electric field for each of the three components on the grid. Constants required in the expression can be set using my_constants. For example, if warpx.Ex_external_grid_function(x,y,z)=Eo*x + delta*(y + z) then the constants Bo and delta required in the above equation can be set using my_constants.Eo= and my_constants.delta= in the input file. For a two-dimensional simulation, it is assumed that the first dimension is x and the second dimension is z, and the value of y is set to zero. Note that the current implementation of the parser for external E-field does not work with RZ and the code will abort with an error message.

  If E_ext_grid_init_style is set to be read_from_file, an additional parameter, indicating the path of an openPMD data file, warpx.read_fields_from_path must be specified, from which external E field data can be loaded into WarpX. One can refer to input files in Examples/Tests/LoadExternalField for more information. Regarding how to prepare the openPMD data file, one can refer to the openPMD-example-datasets. Note that if both B_ext_grid_init_style and E_ext_grid_init_style are set to read_from_file, the openPMD file specified by warpx.read_fields_from_path should contain both B and E external fields data.

• warpx.E_external_grid & warpx.B_external_grid (list of 3 floats)

  required when warpx.E_ext_grid_init_style="constant" and when warpx.B_ext_grid_init_style="constant", respectively. External uniform and constant electrostatic and magnetostatic field added to the grid at initialization. Use with caution as these fields are used for the field solver. In particular, do not use any other boundary condition than periodic.

• warpx.maxlevel_extEMfield_init (default is maximum number of levels in the simulation)

  With this parameter, the externally applied electric and magnetic fields will not be applied for levels greater than warpx.maxlevel_extEMfield_init. For some mesh-refinement simulations, the external fields are only applied to the parent grid and not the refined patches. In such cases, warpx.maxlevel_extEMfield_init can be set to 0. In that case, the other levels have external field values of 0.

Applied to Particles

The external fields defined with input parameters that start with warpx.B_ext_particle_init_ or warpx.E_ext_particle_init_ are applied to the particles directly, at each timestep. As a results, these fields cannot be seen in the diagnostics that output the fields on the grid.

• particles.E_ext_particle_init_style & particles.B_ext_particle_init_style (string) optional (default “none”)

  These parameters determine the type of the external electric and magnetic fields respectively that are applied directly to the particles at every timestep. The field values are specified in the lab frame. With the default none style, no field is applied. Possible values are constant, parse_E_ext_particle_function or parse_B_ext_particle_function, or repeated_plasma_lens.

  • constant: a constant field is applied, given by the input parameters particles.E_external_particle or particles.B_external_particle, which are lists of the field components.

  • parse_E_ext_particle_function or parse_B_ext_particle_function: the field is specified as an analytic expression that is a function of space (x,y,z) and time (t), relative to the lab frame. The E-field is specified by the input parameters:

    • particles.Ex_external_particle_function(x,y,z,t)

    • particles.Ey_external_particle_function(x,y,z,t)

    • particles.Ez_external_particle_function(x,y,z,t)

    The B-field is specified by the input parameters:

    • particles.Bx_external_particle_function(x,y,z,t)

    • particles.By_external_particle_function(x,y,z,t)

    • particles.Bz_external_particle_function(x,y,z,t)

    Note that the position is defined in Cartesian coordinates, as a function of (x,y,z), even for RZ.

  • repeated_plasma_lens: apply a series of plasma lenses. The properties of the lenses are defined in the lab frame by the input parameters:

    • repeated_plasma_lens_period, the period length of the repeat, a single float number,

    • repeated_plasma_lens_starts, the start of each lens relative to the period, an array of floats,

    • repeated_plasma_lens_lengths, the length of each lens, an array of floats,

    • repeated_plasma_lens_strengths_E, the electric focusing strength of each lens, an array of floats, when particles.E_ext_particle_init_style is set to repeated_plasma_lens.

    • repeated_plasma_lens_strengths_B, the magnetic focusing strength of each lens, an array of floats, when particles.B_ext_particle_init_style is set to repeated_plasma_lens.

    The repeated lenses are only defined for \(z > 0\). Once the number of lenses specified in the input are exceeded, the repeated lens stops.

    The applied field is uniform longitudinally (along z) with a hard edge, where residence corrections are used for more accurate field calculation. On the time step when a particle enters or leaves each lens, the field applied is scaled by the fraction of the time step spent within the lens. The fields are of the form \(E_x = \mathrm{strength} \cdot x\), \(E_y = \mathrm{strength} \cdot y\), and \(E_z = 0\), and \(B_x = \mathrm{strength} \cdot y\), \(B_y = -\mathrm{strength} \cdot x\), and \(B_z = 0\).

Applied to Cold Relativistic Fluids

The external fields defined with input parameters that start with warpx.B_ext_init_ or warpx.E_ext_init_ are applied to the fluids directly, at each timestep. As a results, these fields cannot be seen in the diagnostics that output the fields on the grid.

• <fluid_species_name>.E_ext_init_style & <fluid_species_name>.B_ext_init_style (string) optional (default “none”)

  These parameters determine the type of the external electric and magnetic fields respectively that are applied directly to the cold relativistic fluids at every timestep. The field values are specified in the lab frame. With the default none style, no field is applied. Possible values are parse_E_ext_function or parse_B_ext_function.

  • parse_E_ext_function or parse_B_ext_function: the field is specified as an analytic expression that is a function of space (x,y,z) and time (t), relative to the lab frame. The E-field is specified by the input parameters:

    • <fluid_species_name>.Ex_external_function(x,y,z,t)

    • <fluid_species_name>.Ey_external_function(x,y,z,t)

    • <fluid_species_name>.Ez_external_function(x,y,z,t)

    The B-field is specified by the input parameters:

    • <fluid_species_name>.Bx_external_function(x,y,z,t)

    • <fluid_species_name>.By_external_function(x,y,z,t)

    • <fluid_species_name>.Bz_external_function(x,y,z,t)

    Note that the position is defined in Cartesian coordinates, as a function of (x,y,z), even for RZ.

Accelerator Lattice

Several accelerator lattice elements can be defined as described below. The elements are defined relative to the z axis and in the lab frame, starting at z = 0. They are described using a simplified MAD like syntax. Note that elements of the same type cannot overlap each other.

• lattice.elements (list of strings) optional (default: no elements)

  A list of names (one name per lattice element), in the order that they appear in the lattice.

• lattice.reverse (boolean) optional (default: false)

  Reverse the list of elements in the lattice.

• <element_name>.type (string)

  Indicates the element type for this lattice element. This should be one of:

  • drift for free drift. This requires this additional parameter:

    • <element_name>.ds (float, in meters) the segment length

  • quad for a hard edged quadrupole. This applies a quadrupole field that is uniform within the z extent of the element with a sharp cut off at the ends. This uses residence corrections, with the field scaled by the amount of time within the element for particles entering or leaving it, to increase the accuracy. This requires these additional parameters:

    • <element_name>.ds (float, in meters) the segment length

    • <element_name>.dEdx (float, in volts/meter^2) optional (default: 0.) the electric quadrupole field gradient The field applied to the particles will be Ex = dEdx*x and Ey = -dEdx*y.

    • <element_name>.dBdx (float, in Tesla/meter) optional (default: 0.) the magnetic quadrupole field gradient The field applied to the particles will be Bx = dBdx*y and By = dBdx*x.

  • plasmalens for a field modeling a plasma lens This applies a radially directed plasma lens field that is uniform within the z extent of the element with a sharp cut off at the ends. This uses residence corrections, with the field scaled by the amount of time within the element for particles entering or leaving it, to increase the accuracy. This requires these additional parameters:

    • <element_name>.ds (float, in meters) the segment length

    • <element_name>.dEdx (float, in volts/meter^2) optional (default: 0.) the electric field gradient The field applied to the particles will be Ex = dEdx*x and Ey = dEdx*y.

    • <element_name>.dBdx (float, in Tesla/meter) optional (default: 0.) the magnetic field gradient The field applied to the particles will be Bx = dBdx*y and By = -dBdx*x.

  • line a sub-lattice (line) of elements to append to the lattice.

    • <element_name>.elements (list of strings) optional (default: no elements) A list of names (one name per lattice element), in the order that they appear in the lattice.

    • <element_name>.reverse (boolean) optional (default: false) Reverse the list of elements in the line before appending to the lattice.

Collision models

WarpX provides several particle collision models, using varying degrees of approximation. Details about the collision models can be found in the theory section.

• collisions.collision_names (strings, separated by spaces)

  The name of each collision type. This is then used in the rest of the input deck; in this documentation we use <collision_name> as a placeholder.

• <collision_name>.type (string) optional

  The type of collision. The types implemented are:

  • pairwisecoulomb for pair-wise Coulomb collisions, the default if unspecified. This provides a pair-wise relativistic elastic Monte Carlo binary Coulomb collision model, following the algorithm given by Pérez et al. [6]. When the RZ mode is used, warpx.n_rz_azimuthal_modes must be set to 1 at the moment, since the current implementation of the collision module assumes axisymmetry.

  • nuclearfusion for fusion reactions. This implements the pair-wise fusion model by Higginson et al. [7]. Currently, WarpX supports deuterium-deuterium, deuterium-tritium, deuterium-helium and proton-boron fusion. When initializing the reactant and product species, you need to use species_type (see the documentation for this parameter), so that WarpX can identify the type of reaction to use. (e.g. <species_name>.species_type = 'deuterium')

  • dsmc for pair-wise, non-Coulomb collisions between kinetic species. This is a “direct simulation Monte Carlo” treatment of collisions between kinetic species. See DSMC section.

  • background_mcc for collisions between particles and a neutral background. This is a relativistic Monte Carlo treatment for particles colliding with a neutral background gas. See MCC section.

  • background_stopping for slowing of ions due to collisions with electrons or ions. This implements the approximate formulae as derived in Introduction to Plasma Physics, from Goldston and Rutherford, section 14.2.

• <collision_name>.species (strings)

  If using dsmc, pairwisecoulomb or nuclearfusion, this should be the name(s) of the species, between which the collision will be considered. (Provide only one name for intra-species collisions.) If using background_mcc or background_stopping type this should be the name of the species for which collisions with a background will be included. In this case, only one species name should be given.

• <collision_name>.product_species (strings)

  Only for nuclearfusion. The name(s) of the species in which to add the new macroparticles created by the reaction.

• <collision_name>.ndt (int) optional

  Execute collision every # time steps. The default value is 1.

• <collision_name>.CoulombLog (float) optional

  Only for pairwisecoulomb. A provided fixed Coulomb logarithm of the collision type <collision_name>. For example, a typical Coulomb logarithm has a form of \(\ln(\lambda_D/R)\), where \(\lambda_D\) is the Debye length, \(R\approx1.4A^{1/3}\) is the effective Coulombic radius of the nucleus, \(A\) is the mass number. If this is not provided, or if a non-positive value is provided, a Coulomb logarithm will be computed automatically according to the algorithm in Pérez et al. [6].

• <collision_name>.fusion_multiplier (float) optional.

  Only for nuclearfusion. Increasing fusion_multiplier creates more macroparticles of fusion products, but with lower weight (in such a way that the corresponding total number of physical particle remains the same). This can improve the statistics of the simulation, in the case where fusion reactions are very rare. More specifically, in a fusion reaction between two macroparticles with weight w_1 and w_2, the weight of the product macroparticles will be min(w_1,w_2)/fusion_multiplier. (And the weights of the reactant macroparticles are reduced correspondingly after the reaction.) See Higginson et al. [7] for more details. The default value of fusion_multiplier is 1.

• <collision_name>.fusion_probability_threshold (float) optional.

  Only for nuclearfusion. If the fusion multiplier is too high and results in a fusion probability that approaches 1 (for a given collision between two macroparticles), then there is a risk of underestimating the total fusion yield. In these cases, WarpX reduces the fusion multiplier used in that given collision. m_probability_threshold is the fusion probability threshold above which WarpX reduces the fusion multiplier.

• <collision_name>.fusion_probability_target_value (float) optional.

  Only for nuclearfusion. When the probability of fusion for a given collision exceeds fusion_probability_threshold, WarpX reduces the fusion multiplier for that collisions such that the fusion probability approches fusion_probability_target_value.

• <collision_name>.background_density (float)

  Only for background_mcc and background_stopping. The density of the background in \(m^{-3}\). Can also provide <collision_name>.background_density(x,y,z,t) using the parser initialization style for spatially and temporally varying density. With background_mcc, if a function is used for the background density, the input parameter <collision_name>.max_background_density must also be provided to calculate the maximum collision probability.

• <collision_name>.background_temperature (float)

  Only for background_mcc and background_stopping. The temperature of the background in Kelvin. Can also provide <collision_name>.background_temperature(x,y,z,t) using the parser initialization style for spatially and temporally varying temperature.

• <collision_name>.background_mass (float) optional

  Only for background_mcc and background_stopping. The mass of the background gas in kg. With background_mcc, if not given the mass of the colliding species will be used unless ionization is included in which case the mass of the product species will be used. With background_stopping, and background_type set to electrons, if not given defaults to the electron mass. With background_type set to ions, the mass must be given.

• <collision_name>.background_charge_state (float)

  Only for background_stopping, where it is required when background_type is set to ions. This specifies the charge state of the background ions.

• <collision_name>.background_type (string)

  Only for background_stopping, where it is required, the type of the background. The possible values are electrons and ions. When electrons, equation 14.12 from Goldston and Rutherford is used. This formula is based on Coulomb collisions with the approximations that \(M_b >> m_e\) and \(V << v_{thermal\_e}\), and the assumption that the electrons have a Maxwellian distribution with temperature \(T_e\).

  \[\frac{dV}{dt} = - \frac{2^{1/2}n_eZ_b^2e^4m_e^{1/2}\log\Lambda}{12\pi^{3/2}\epsilon_0M_bT_e^{3/2}}V\]

  where \(V\) is each velocity component, \(n_e\) is the background density, \(Z_b\) is the ion charge state, \(e\) is the electron charge, \(m_e\) is the background mass, \(\log\Lambda=\log((12\pi/Z_b)(n_e\lambda_{de}^3))\), \(\lambda_{de}\) is the DeBye length, and \(M_b\) is the ion mass. The equation is integrated over a time step, giving \(V(t+dt) = V(t)*\exp(-\alpha*{dt})\) where \(\alpha\) is the factor multiplying \(V\).

  When ions, equation 14.20 is used. This formula is based on Coulomb collisions with the approximations that \(M_b >> M\) and \(V >> v_{thermal\_i}\). The background ion temperature only appears in the \(\log\Lambda\) term.

  \[\frac{dW_b}{dt} = - \frac{2^{1/2}n_iZ^2Z_b^2e^4M_b^{1/2}\log\Lambda}{8\pi\epsilon_0MW_b^{1/2}}\]

  where \(W_b\) is the ion energy, \(n_i\) is the background density, \(Z\) is the charge state of the background ions, \(Z_b\) is the ion charge state, \(e\) is the electron charge, \(M_b\) is the ion mass, \(\log\Lambda=\log((12\pi/Z_b)(n_i\lambda_{di}^3))\), \(\lambda_{di}\) is the DeBye length, and \(M\) is the background ion mass. The equation is integrated over a time step, giving \(W_b(t+dt) = ((W_b(t)^{3/2}) - 3/2\beta{dt})^{2/3}\) where \(\beta\) is the term on the r.h.s except \(W_b\).

• <collision_name>.scattering_processes (strings separated by spaces)

  Only for dsmc and background_mcc. The scattering processes that should be included. Available options are elastic, back & charge_exchange for ions and elastic, excitationX & ionization for electrons. Multiple excitation events can be included for electrons corresponding to excitation to different levels, the X above can be changed to a unique identifier for each excitation process. For each scattering process specified a path to a cross-section data file must also be given. We use <scattering_process> as a placeholder going forward.

• <collision_name>.<scattering_process>_cross_section (string)

  Only for dsmc and background_mcc. Path to the file containing cross-section data for the given scattering processes. The cross-section file must have exactly 2 columns of data, the first containing equally spaced energies in eV and the second the corresponding cross-section in \(m^2\). The energy column should represent the kinetic energy of the colliding particles in the center-of-mass frame.

• <collision_name>.<scattering_process>_energy (float)

  Only for background_mcc. If the scattering process is either excitationX or ionization the energy cost of that process must be given in eV.

• <collision_name>.ionization_species (float)

  Only for background_mcc. If the scattering process is ionization the produced species must also be given. For example if argon properties is used for the background gas, a species of argon ions should be specified here.

Numerics and algorithms

This section describes the input parameters used to select numerical methods and algorithms for your simulation setup.

Time step

• warpx.cfl (float) optional (default 0.999)

  The ratio between the actual timestep that is used in the simulation and the Courant-Friedrichs-Lewy (CFL) limit. (e.g. for warpx.cfl=1, the timestep will be exactly equal to the CFL limit.) This parameter will only be used with the electromagnetic solver.

• warpx.const_dt (float)

  Allows direct specification of the time step size, in units of seconds. When the electrostatic solver is being used, this must be supplied. This can be used with the electromagnetic solver, overriding warpx.cfl, but it is up to the user to ensure that the CFL condition is met.

Filtering

• warpx.use_filter (0 or 1; default: 1, except for RZ FDTD)

  Whether to smooth the charge and currents on the mesh, after depositing them from the macro-particles. This uses a bilinear filter (see the filtering section). The default is 1 in all cases, except for simulations in RZ geometry using the FDTD solver. With the RZ PSATD solver, the filtering is done in \(k\)-space.

  Warning

  Known bug: filter currently not working with FDTD solver in RZ geometry (see https://github.com/ECP-WarpX/WarpX/issues/1943).

• warpx.filter_npass_each_dir (3 int) optional (default 1 1 1)

  Number of passes along each direction for the bilinear filter. In 2D simulations, only the first two values are read.

• warpx.use_filter_compensation (0 or 1; default: 0)

  Whether to add compensation when applying filtering. This is only supported with the RZ spectral solver.

Particle push, charge and current deposition, field gathering

• algo.current_deposition (string, optional)

  This parameter selects the algorithm for the deposition of the current density. Available options are: direct, esirkepov, and vay. The default choice is esirkepov for FDTD maxwell solvers but direct for standard or Galilean PSATD solver (i.e. with algo.maxwell_solver = psatd) and for the hybrid-PIC solver (i.e. with algo.maxwell_solver = hybrid) and for diagnostics output with the electrostatic solvers (i.e., with warpx.do_electrostatic = ...). Note that vay is only available for algo.maxwell_solver = psatd.

  1. direct

     The current density is deposited as described in the section Current deposition. This deposition scheme does not conserve charge.

  2. esirkepov

     The current density is deposited as described in Esirkepov [8]. This deposition scheme guarantees charge conservation for shape factors of arbitrary order.

  3. vay

     The current density is deposited as described in Vay et al. [9] (see section Current deposition for more details). This option guarantees charge conservation only when used in combination with psatd.periodic_single_box_fft=1, that is, only for periodic single-box simulations with global FFTs without guard cells. The implementation for domain decomposition with local FFTs over guard cells is planned but not yet completed.

• algo.charge_deposition (string, optional)

  The algorithm for the charge density deposition. Available options are:

  • standard: standard charge deposition algorithm, described in the particle-in-cell theory section.

• algo.field_gathering (string, optional)

  The algorithm for field gathering. Available options are:

  • energy-conserving: gathers directly from the grid points (either staggered or nodal grid points depending on warpx.grid_type).

  • momentum-conserving: first average the fields from the grid points to the nodes, and then gather from the nodes.

  Default: algo.field_gathering = energy-conserving with collocated or staggered grids (note that energy-conserving and momentum-conserving are equivalent with collocated grids), algo.field_gathering = momentum-conserving with hybrid grids.

• algo.particle_pusher (string, optional)

  The algorithm for the particle pusher. Available options are:

  • boris: Boris pusher.

  • vay: Vay pusher (see Vay [1])

  • higuera: Higuera-Cary pusher (see Higuera and Cary [10])

  If algo.particle_pusher is not specified, boris is the default.

• algo.particle_shape (integer; 1, 2, 3, or 4)

  The order of the shape factors (splines) for the macro-particles along all spatial directions: 1 for linear, 2 for quadratic, 3 for cubic, 4 for quartic. Low-order shape factors result in faster simulations, but may lead to more noisy results. High-order shape factors are computationally more expensive, but may increase the overall accuracy of the results. For production runs it is generally safer to use high-order shape factors, such as cubic order.

  Note that this input parameter is not optional and must always be set in all input files provided that there is at least one particle species (set in input as particles.species_names) or one laser species (set in input as lasers.names) in the simulation. No default value is provided automatically.

Maxwell solver

Two families of Maxwell solvers are implemented in WarpX, based on the Finite-Difference Time-Domain method (FDTD) or the Pseudo-Spectral Analytical Time-Domain method (PSATD), respectively.

• algo.maxwell_solver (string, optional)

  The algorithm for the Maxwell field solver. Available options are:

  • yee: Yee FDTD solver.

  • ckc: (not available in RZ geometry) Cole-Karkkainen solver with Cowan coefficients (see Cowan et al. [11]).

  • psatd: Pseudo-spectral solver (see theory).

  • ect: Enlarged cell technique (conformal finite difference solver. See Xiao and Liu [12]).

  • hybrid: The E-field will be solved using Ohm’s law and a kinetic-fluid hybrid model (see theory).

  • none: No field solve will be performed.

  If algo.maxwell_solver is not specified, yee is the default.

• algo.em_solver_medium (string, optional)

  The medium for evaluating the Maxwell solver. Available options are :

  • vacuum: vacuum properties are used in the Maxwell solver.

  • macroscopic: macroscopic Maxwell equation is evaluated. If this option is selected, then the corresponding properties of the medium must be provided using macroscopic.sigma, macroscopic.epsilon, and macroscopic.mu for each case where the initialization style is constant. Otherwise if the initialization style uses the parser, macroscopic.sigma_function(x,y,z), macroscopic.epsilon_function(x,y,z) and/or macroscopic.mu_function(x,y,z) must be provided using the parser initialization style for spatially varying macroscopic properties.

  If algo.em_solver_medium is not specified, vacuum is the default.

Maxwell solver: PSATD method

• psatd.nox, psatd.noy, pstad.noz (integer) optional (default 16 for all)

  The order of accuracy of the spatial derivatives, when using the code compiled with a PSATD solver. If psatd.periodic_single_box_fft is used, these can be set to inf for infinite-order PSATD.

• psatd.nx_guard, psatd.ny_guard, psatd.nz_guard (integer) optional

  The number of guard cells to use with PSATD solver. If not set by users, these values are calculated automatically and determined empirically and equal the order of the solver for collocated grids and half the order of the solver for staggered grids.

• psatd.periodic_single_box_fft (0 or 1; default: 0)

  If true, this will not incorporate the guard cells into the box over which FFTs are performed. This is only valid when WarpX is run with periodic boundaries and a single box. In this case, using psatd.periodic_single_box_fft is equivalent to using a global FFT over the whole domain. Therefore, all the approximations that are usually made when using local FFTs with guard cells (for problems with multiple boxes) become exact in the case of the periodic, single-box FFT without guard cells.

• psatd.current_correction (0 or 1; default: 1, with the exceptions mentioned below)

  If true, a current correction scheme in Fourier space is applied in order to guarantee charge conservation. The default value is psatd.current_correction=1, unless a charge-conserving current deposition scheme is used (by setting algo.current_deposition=esirkepov or algo.current_deposition=vay) or unless the div(E) cleaning scheme is used (by setting warpx.do_dive_cleaning=1).

  If psatd.v_galilean is zero, the spectral solver used is the standard PSATD scheme described in Vay et al. [9] and the current correction reads

  \[\widehat{\boldsymbol{J}}^{\,n+1/2}_{\mathrm{correct}} = \widehat{\boldsymbol{J}}^{\,n+1/2} - \bigg(\boldsymbol{k}\cdot\widehat{\boldsymbol{J}}^{\,n+1/2} - i \frac{\widehat{\rho}^{n+1} - \widehat{\rho}^{n}}{\Delta{t}}\bigg) \frac{\boldsymbol{k}}{k^2}\]

  If psatd.v_galilean is non-zero, the spectral solver used is the Galilean PSATD scheme described in Lehe et al. [13] and the current correction reads

  \[\widehat{\boldsymbol{J}}^{\,n+1/2}_{\mathrm{correct}} = \widehat{\boldsymbol{J}}^{\,n+1/2} - \bigg(\boldsymbol{k}\cdot\widehat{\boldsymbol{J}}^{\,n+1/2} - (\boldsymbol{k}\cdot\boldsymbol{v}_G) \,\frac{\widehat\rho^{n+1} - \widehat\rho^{n}\theta^2}{1 - \theta^2}\bigg) \frac{\boldsymbol{k}}{k^2}\]

  where \(\theta=\exp(i\,\boldsymbol{k}\cdot\boldsymbol{v}_G\,\Delta{t}/2)\).

  This option is currently implemented only for the standard PSATD, Galilean PSATD, and averaged Galilean PSATD schemes, while it is not yet available for the multi-J algorithm.

• psatd.update_with_rho (0 or 1)

  If true, the update equation for the electric field is expressed in terms of both the current density and the charge density, namely \(\widehat{\boldsymbol{J}}^{\,n+1/2}\), \(\widehat\rho^{n}\), and \(\widehat\rho^{n+1}\). If false, instead, the update equation for the electric field is expressed in terms of the current density \(\widehat{\boldsymbol{J}}^{\,n+1/2}\) only. If charge is expected to be conserved (by setting, for example, psatd.current_correction=1), then the two formulations are expected to be equivalent.

  If psatd.v_galilean is zero, the spectral solver used is the standard PSATD scheme described in Vay et al. [9]:

  1. if psatd.update_with_rho=0, the update equation for the electric field reads

  \[\begin{split}\begin{split} \widehat{\boldsymbol{E}}^{\,n+1}= & \: C \widehat{\boldsymbol{E}}^{\,n} + i \, \frac{S c}{k} \boldsymbol{k}\times\widehat{\boldsymbol{B}}^{\,n} - \frac{S}{\epsilon_0 c \, k} \widehat{\boldsymbol{J}}^{\,n+1/2} \\[0.2cm] & +\frac{1-C}{k^2} (\boldsymbol{k}\cdot\widehat{\boldsymbol{E}}^{\,n}) \boldsymbol{k} + \frac{1}{\epsilon_0 k^2} \left(\frac{S}{c \, k}-\Delta{t}\right) (\boldsymbol{k}\cdot\widehat{\boldsymbol{J}}^{\,n+1/2}) \boldsymbol{k} \end{split}\end{split}\]

  2. if psatd.update_with_rho=1, the update equation for the electric field reads

  \[\begin{split}\begin{split} \widehat{\boldsymbol{E}}^{\,n+1}= & \: C\widehat{\boldsymbol{E}}^{\,n} + i \, \frac{S c}{k} \boldsymbol{k}\times\widehat{\boldsymbol{B}}^{\,n} - \frac{S}{\epsilon_0 c \, k} \widehat{\boldsymbol{J}}^{\,n+1/2} \\[0.2cm] & + \frac{i}{\epsilon_0 k^2} \left(C-\frac{S}{c\,k}\frac{1}{\Delta{t}}\right) \widehat{\rho}^{n} \boldsymbol{k} - \frac{i}{\epsilon_0 k^2} \left(1-\frac{S}{c \, k} \frac{1}{\Delta{t}}\right)\widehat{\rho}^{n+1} \boldsymbol{k} \end{split}\end{split}\]

  The coefficients \(C\) and \(S\) are defined in Vay et al. [9].

  If psatd.v_galilean is non-zero, the spectral solver used is the Galilean PSATD scheme described in Lehe et al. [13]:

  1. if psatd.update_with_rho=0, the update equation for the electric field reads

  \[\begin{split}\begin{split} \widehat{\boldsymbol{E}}^{\,n+1} = & \: \theta^{2} C \widehat{\boldsymbol{E}}^{\,n} + i \, \theta^{2} \frac{S c}{k} \boldsymbol{k}\times\widehat{\boldsymbol{B}}^{\,n} + \frac{i \, \nu \, \theta \, \chi_1 - \theta^{2} S}{\epsilon_0 c \, k} \widehat{\boldsymbol{J}}^{\,n+1/2} \\[0.2cm] & + \theta^{2} \frac{\chi_2-\chi_3}{k^{2}} (\boldsymbol{k}\cdot\widehat{\boldsymbol{E}}^{\,n}) \boldsymbol{k} + i \, \frac{\chi_2\left(\theta^{2}-1\right)}{\epsilon_0 c \, k^{3} \nu} (\boldsymbol{k}\cdot\widehat{\boldsymbol{J}}^{\,n+1/2}) \boldsymbol{k} \end{split}\end{split}\]

  2. if psatd.update_with_rho=1, the update equation for the electric field reads

  \[\begin{split}\begin{split} \widehat{\boldsymbol{E}}^{\,n+1} = & \: \theta^{2} C \widehat{\boldsymbol{E}}^{\,n} + i \, \theta^{2} \frac{S c}{k} \boldsymbol{k}\times\widehat{\boldsymbol{B}}^{\,n} + \frac{i \, \nu \, \theta \, \chi_1 - \theta^{2} S}{\epsilon_0 c \, k} \widehat{\boldsymbol{J}}^{\,n+1/2} \\[0.2cm] & + i \, \frac{\theta^{2} \chi_3}{\epsilon_0 k^{2}} \widehat{\rho}^{\,n} \boldsymbol{k} - i \, \frac{\chi_2}{\epsilon_0 k^{2}} \widehat{\rho}^{\,n+1} \boldsymbol{k} \end{split}\end{split}\]

  The coefficients \(C\), \(S\), \(\theta\), \(\nu\), \(\chi_1\), \(\chi_2\), and \(\chi_3\) are defined in Lehe et al. [13].

  The default value for psatd.update_with_rho is 1 if psatd.v_galilean is non-zero and 0 otherwise. The option psatd.update_with_rho=0 is not implemented with the following algorithms: comoving PSATD (psatd.v_comoving), time averaging (psatd.do_time_averaging=1), div(E) cleaning (warpx.do_dive_cleaning=1), and multi-J (warpx.do_multi_J=1).

  Note that the update with and without rho is also supported in RZ geometry.

• psatd.J_in_time (constant or linear; default constant)

  This determines whether the current density is assumed to be constant or linear in time, within the time step over which the electromagnetic fields are evolved.

• psatd.rho_in_time (linear; default linear)

  This determines whether the charge density is assumed to be linear in time, within the time step over which the electromagnetic fields are evolved.

• psatd.v_galilean (3 floats, in units of the speed of light; default 0. 0. 0.)

  Defines the Galilean velocity. A non-zero velocity activates the Galilean algorithm, which suppresses numerical Cherenkov instabilities (NCI) in boosted-frame simulations (see the section Numerical Stability and alternate formulation in a Galilean frame for more information). This requires the code to be compiled with the spectral solver. It also requires the use of the direct current deposition algorithm (by setting algo.current_deposition = direct).

• psatd.use_default_v_galilean (0 or 1; default: 0)

  This can be used in boosted-frame simulations only and sets the Galilean velocity along the \(z\) direction automatically as \(v_{G} = -\sqrt{1-1/\gamma^2}\), where \(\gamma\) is the Lorentz factor of the boosted frame (set by warpx.gamma_boost). See the section Numerical Stability and alternate formulation in a Galilean frame for more information on the Galilean algorithm for boosted-frame simulations.

• psatd.v_comoving (3 floating-point values, in units of the speed of light; default 0. 0. 0.)

  Defines the comoving velocity in the comoving PSATD scheme. A non-zero comoving velocity selects the comoving PSATD algorithm, which suppresses the numerical Cherenkov instability (NCI) in boosted-frame simulations, under certain assumptions. This option requires that WarpX is compiled with USE_PSATD = TRUE. It also requires the use of direct current deposition (algo.current_deposition = direct) and has not been neither implemented nor tested with other current deposition schemes.

• psatd.do_time_averaging (0 or 1; default: 0)

  Whether to use an averaged Galilean PSATD algorithm or standard Galilean PSATD.

• warpx.do_multi_J (0 or 1; default: 0)

  Whether to use the multi-J algorithm, where current deposition and field update are performed multiple times within each time step. The number of sub-steps is determined by the input parameter warpx.do_multi_J_n_depositions. Unlike sub-cycling, field gathering is performed only once per time step, as in regular PIC cycles. When warpx.do_multi_J = 1, we perform linear interpolation of two distinct currents deposited at the beginning and the end of the time step, instead of using one single current deposited at half time. For simulations with strong numerical Cherenkov instability (NCI), it is recommended to use the multi-J algorithm in combination with psatd.do_time_averaging = 1.

• warpx.do_multi_J_n_depositions (integer)

  Number of sub-steps to use with the multi-J algorithm, when warpx.do_multi_J = 1. Note that this input parameter is not optional and must always be set in all input files where warpx.do_multi_J = 1. No default value is provided automatically.

Maxwell solver: macroscopic media

• algo.macroscopic_sigma_method (string, optional)

  The algorithm for updating electric field when algo.em_solver_medium is macroscopic. Available options are:

  • backwardeuler is a fully-implicit, first-order in time scheme for E-update (default).

  • laxwendroff is the semi-implicit, second order in time scheme for E-update.

  Comparing the two methods, Lax-Wendroff is more prone to developing oscillations and requires a smaller timestep for stability. On the other hand, Backward Euler is more robust but it is first-order accurate in time compared to the second-order Lax-Wendroff method.

• macroscopic.sigma_function(x,y,z), macroscopic.epsilon_function(x,y,z), macroscopic.mu_function(x,y,z) (string)

  To initialize spatially varying conductivity, permittivity, and permeability, respectively, using a mathematical function in the input. Constants required in the mathematical expression can be set using my_constants. These parameters are parsed if algo.em_solver_medium=macroscopic.

• macroscopic.sigma, macroscopic.epsilon, macroscopic.mu (double)

  To initialize a constant conductivity, permittivity, and permeability of the computational medium, respectively. The default values are the corresponding values in vacuum.

Maxwell solver: kinetic-fluid hybrid

• hybrid_pic_model.elec_temp (float)

  If algo.maxwell_solver is set to hybrid, this sets the electron temperature, in eV, used to calculate the electron pressure (see here).

• hybrid_pic_model.n0_ref (float)

  If algo.maxwell_solver is set to hybrid, this sets the reference density, in \(m^{-3}\), used to calculate the electron pressure (see here).

• hybrid_pic_model.gamma (float) optional (default 5/3)

  If algo.maxwell_solver is set to hybrid, this sets the exponent used to calculate the electron pressure (see here).

• hybrid_pic_model.plasma_resistivity(rho,J) (float or str) optional (default 0)

  If algo.maxwell_solver is set to hybrid, this sets the plasma resistivity in \(\Omega m\).

• hybrid_pic_model.plasma_hyper_resistivity (float or str) optional (default 0)

  If algo.maxwell_solver is set to hybrid, this sets the plasma hyper-resistivity in \(\Omega m^3\).

• hybrid_pic_model.J[x/y/z]_external_grid_function(x, y, z, t) (float or str) optional (default 0)

  If algo.maxwell_solver is set to hybrid, this sets the external current (on the grid) in \(A/m^2\).

• hybrid_pic_model.n_floor (float) optional (default 1)

  If algo.maxwell_solver is set to hybrid, this sets the plasma density floor, in \(m^{-3}\), which is useful since the generalized Ohm’s law used to calculate the E-field includes a \(1/n\) term.

• hybrid_pic_model.substeps (int) optional (default 10)

  If algo.maxwell_solver is set to hybrid, this sets the number of sub-steps to take during the B-field update.

Note

Based on results from Stanier et al. [14] it is recommended to use linear particles when using the hybrid-PIC model.

Grid types (collocated, staggered, hybrid)

• warpx.grid_type (string, collocated, staggered or hybrid)

  Whether to use a collocated grid (all fields defined at the cell nodes), a staggered grid (fields defined on a Yee grid), or a hybrid grid (fields and currents are interpolated back and forth between a staggered grid and a nodal grid, must be used with momentum-conserving field gathering algorithm, algo.field_gathering = momentum-conserving). The option hybrid is currently not supported in RZ geometry.

  Default: warpx.grid_type = staggered.

• interpolation.galerkin_scheme (0 or 1)

  Whether to use a Galerkin scheme when gathering fields to particles. When set to 1, the interpolation orders used for field-gathering are reduced for certain field components along certain directions. For example, \(E_z\) is gathered using algo.particle_shape along \((x,y)\) and algo.particle_shape - 1 along \(z\). See equations (21)-(23) of Godfrey and Vay [15] and associated references for details.

  Default: interpolation.galerkin_scheme = 0 with collocated grids and/or momentum-conserving field gathering, interpolation.galerkin_scheme = 1 otherwise.

  Warning

  The default behavior should not normally be changed. At present, this parameter is intended mainly for testing and development purposes.

• warpx.field_centering_nox, warpx.field_centering_noy, warpx.field_centering_noz (integer, optional)

  The order of interpolation used with staggered or hybrid grids (warpx.grid_type = staggered or warpx.grid_type = hybrid) and momentum-conserving field gathering (algo.field_gathering = momentum-conserving) to interpolate the electric and magnetic fields from the cell centers to the cell nodes, before gathering the fields from the cell nodes to the particle positions.

  Default: warpx.field_centering_no<x,y,z> = 2 with staggered grids, warpx.field_centering_no<x,y,z> = 8 with hybrid grids (typically necessary to ensure stability in boosted-frame simulations of relativistic plasmas and beams).

• warpx.current_centering_nox, warpx.current_centering_noy, warpx.current_centering_noz (integer, optional)

  The order of interpolation used with hybrid grids (warpx.grid_type = hybrid) to interpolate the currents from the cell nodes to the cell centers when warpx.do_current_centering = 1, before pushing the Maxwell fields on staggered grids.

  Default: warpx.current_centering_no<x,y,z> = 8 with hybrid grids (typically necessary to ensure stability in boosted-frame simulations of relativistic plasmas and beams).

• warpx.do_current_centering (bool, 0 or 1)

  If true, the current is deposited on a nodal grid and then centered to a staggered grid (Yee grid), using finite-order interpolation.

  Default: warpx.do_current_centering = 0 with collocated or staggered grids, warpx.do_current_centering = 1 with hybrid grids.

Additional parameters

• warpx.do_dive_cleaning (0 or 1 ; default: 0)

  Whether to use modified Maxwell equations that progressively eliminate the error in \(div(E)-\rho\). This can be useful when using a current deposition algorithm which is not strictly charge-conserving, or when using mesh refinement. These modified Maxwell equation will cause the error to propagate (at the speed of light) to the boundaries of the simulation domain, where it can be absorbed.

• warpx.do_subcycling (0 or 1; default: 0)

  Whether or not to use sub-cycling. Different refinement levels have a different cell size, which results in different Courant–Friedrichs–Lewy (CFL) limits for the time step. By default, when using mesh refinement, the same time step is used for all levels. This time step is taken as the CFL limit of the finest level. Hence, for coarser levels, the timestep is only a fraction of the CFL limit for this level, which may lead to numerical artifacts. With sub-cycling, each level evolves with its own time step, set to its own CFL limit. In practice, it means that when level 0 performs one iteration, level 1 performs two iterations. Currently, this option is only supported when amr.max_level = 1. More information can be found at https://ieeexplore.ieee.org/document/8659392.

• warpx.override_sync_intervals (string) optional (default 1)

  Using the Intervals parser syntax, this string defines the timesteps at which synchronization of sources (rho and J) and fields (E and B) on grid nodes at box boundaries is performed. Since the grid nodes at the interface between two neighbor boxes are duplicated in both boxes, an instability can occur if they have too different values. This option makes sure that they are synchronized periodically. Note that if Perfectly Matched Layers (PML) are used, synchronization of the E and B fields is performed at every timestep regardless of this parameter.

• warpx.use_hybrid_QED (bool; default: 0)

  Will use the Hybrid QED Maxwell solver when pushing fields: a QED correction is added to the field solver to solve non-linear Maxwell’s equations, according to Grismayer et al. [16]. Note that this option can only be used with the PSATD build. Furthermore, one must set warpx.grid_type = collocated (which otherwise would be staggered by default).

• warpx.quantum_xi (float; default: 1.3050122.e-52)

  Overwrites the actual quantum parameter used in Maxwell’s QED equations. Assigning a value here will make the simulation unphysical, but will allow QED effects to become more apparent. Note that this option will only have an effect if the warpx.use_Hybrid_QED flag is also triggered.

• warpx.do_device_synchronize (bool) optional (default 1)

  When running in an accelerated platform, whether to call a amrex::Gpu::synchronize() around profiling regions. This allows the profiler to give meaningful timers, but (hardly) slows down the simulation.

• warpx.sort_intervals (string) optional (defaults: -1 on CPU; 4 on GPU)

  Using the Intervals parser syntax, this string defines the timesteps at which particles are sorted. If <=0, do not sort particles. It is turned on on GPUs for performance reasons (to improve memory locality).

• warpx.sort_particles_for_deposition (bool) optional (default: true for the CUDA backend, otherwise false)

  This option controls the type of sorting used if particle sorting is turned on, i.e. if sort_intervals is not <=0. If true, particles will be sorted by cell to optimize deposition with many particles per cell, in the order x -> y -> z -> ppc. If false, particles will be sorted by bin, using the sort_bin_size parameter below, in the order ppc -> x -> y -> z. true is recommend for best performance on NVIDIA GPUs, especially if there are many particles per cell.

• warpx.sort_idx_type (list of int) optional (default: 0 0 0)

  This controls the type of grid used to sort the particles when sort_particles_for_deposition is true. Possible values are: idx_type = {0, 0, 0}: Sort particles to a cell centered grid idx_type = {1, 1, 1}: Sort particles to a node centered grid idx_type = {2, 2, 2}: Compromise between a cell and node centered grid. In 2D (XZ and RZ), only the first two elements are read. In 1D, only the first element is read.

• warpx.sort_bin_size (list of int) optional (default 1 1 1)

  If sort_intervals is activated and sort_particles_for_deposition is false, particles are sorted in bins of sort_bin_size cells. In 2D, only the first two elements are read.

• warpx.do_shared_mem_charge_deposition (bool) optional (default false)

  If activated, charge deposition will allocate and use small temporary buffers on which to accumulate deposited charge values from particles. On GPUs these buffers will reside in __shared__ memory, which is faster than the usual __global__ memory. Performance impact will depend on the relative overhead of assigning the particles to bins small enough to fit in the space available for the temporary buffers.

• warpx.do_shared_mem_current_deposition (bool) optional (default false)

  If activated, current deposition will allocate and use small temporary buffers on which to accumulate deposited current values from particles. On GPUs these buffers will reside in __shared__ memory, which is faster than the usual __global__ memory. Performance impact will depend on the relative overhead of assigning the particles to bins small enough to fit in the space available for the temporary buffers. Performance is mostly improved when there is lots of contention between particles writing to the same cell (e.g. for high particles per cell). This feature is only available for CUDA and HIP, and is only recommended for 3D or 2D.

• warpx.shared_tilesize (list of int) optional (default 6 6 8 in 3D; 14 14 in 2D; 1s otherwise)

  Used to tune performance when do_shared_mem_current_deposition or do_shared_mem_charge_deposition is enabled. shared_tilesize is the size of the temporary buffer allocated in shared memory for a threadblock. A larger tilesize requires more shared memory, but gives more work to each threadblock, which can lead to higher occupancy, and allows for more buffered writes to __shared__ instead of __global__. The defaults in 2D and 3D are chosen from experimentation, but can be improved upon for specific problems. The other defaults are not optimized and should always be fine tuned for the problem.

• warpx.shared_mem_current_tpb (int) optional (default 128)

  Used to tune performance when do_shared_mem_current_deposition is enabled. shared_mem_current_tpb controls the number of threads per block (tpb), i.e. the number of threads operating on a shared buffer.

Diagnostics and output

In-situ visualization

WarpX has four types of diagnostics: FullDiagnostics consist in dumps of fields and particles at given iterations, BackTransformedDiagnostics are used when running a simulation in a boosted frame, to reconstruct output data to the lab frame, BoundaryScrapingDiagnostics are used to collect the particles that are absorbed at the boundary, throughout the simulation, and ReducedDiags allow the user to compute some reduced quantity (particle temperature, max of a field) and write a small amount of data to text files. Similar to what is done for physical species, WarpX has a class Diagnostics that allows users to initialize different diagnostics, each of them with different fields, resolution and period. This currently applies to standard diagnostics, but should be extended to back-transformed diagnostics and reduced diagnostics (and others) in a near future.

Full Diagnostics

FullDiagnostics consist in dumps of fields and particles at given iterations. Similar to what is done for physical species, WarpX has a class Diagnostics that allows users to initialize different diagnostics, each of them with different fields, resolution and period. The user specifies the number of diagnostics and the name of each of them, and then specifies options for each of them separately. Note that some parameter (those that do not start with a <diag_name>. prefix) apply to all diagnostics. This should be changed in the future. In-situ capabilities can be used by turning on Sensei or Ascent (provided they are installed) through the output format, see below.

• diagnostics.enable (0 or 1, optional, default 1)

  Whether to enable or disable diagnostics. This flag overwrites all other diagnostics input parameters.

• diagnostics.diags_names (list of string optional, default empty)

  Name of each diagnostics. example: diagnostics.diags_names = diag1 my_second_diag.

• <diag_name>.intervals (string)

  Using the Intervals parser syntax, this string defines the timesteps at which data is dumped. Use a negative number or 0 to disable data dumping. example: diag1.intervals = 10,20:25:1. Note that by default the last timestep is dumped regardless of this parameter. This can be changed using the parameter <diag_name>.dump_last_timestep described below.

• <diag_name>.dump_last_timestep (bool optional, default 1)

  If this is 1, the last timestep is dumped regardless of <diag_name>.intervals.

• <diag_name>.diag_type (string)

  Type of diagnostics. Full, BackTransformed, and BoundaryScraping example: diag1.diag_type = Full or diag1.diag_type = BackTransformed

• <diag_name>.format (string optional, default plotfile)

  Flush format. Possible values are:

  • plotfile for native AMReX format.

  • checkpoint for a checkpoint file, only works with <diag_name>.diag_type = Full.

  • openpmd for OpenPMD format openPMD. Requires to build WarpX with USE_OPENPMD=TRUE (see instructions).

  • ascent for in-situ visualization using Ascent.

  • sensei for in-situ visualization using Sensei.

  example: diag1.format = openpmd.

• <diag_name>.sensei_config (string)

  Only read if <diag_name>.format = sensei. Points to the SENSEI XML file which selects and configures the desired back end.

• <diag_name>.sensei_pin_mesh (integer; 0 by default)

  Only read if <diag_name>.format = sensei. When 1 lower left corner of the mesh is pinned to 0.,0.,0.

• <diag_name>.openpmd_backend (bp, h5 or json) optional, only used if <diag_name>.format = openpmd

  I/O backend for openPMD data dumps. bp is the ADIOS I/O library, h5 is the HDF5 format, and json is a simple text format. json only works with serial/single-rank jobs. When WarpX is compiled with openPMD support, the first available backend in the order given above is taken.

• <diag_name>.openpmd_encoding (optional, v (variable based), f (file based) or g (group based) ) only read if <diag_name>.format = openpmd.

  openPMD file output encoding. File based: one file per timestep (slower), group/variable based: one file for all steps (faster)). variable based is an experimental feature with ADIOS2 and not supported for back-transformed diagnostics. Default: f (full diagnostics)

• <diag_name>.adios2_operator.type (zfp, blosc) optional,

  ADIOS2 I/O operator type for openPMD data dumps.

• <diag_name>.adios2_operator.parameters.* optional,

  ADIOS2 I/O operator parameters for openPMD data dumps.

  A typical example for ADIOS2 output using lossless compression with blosc using the zstd compressor and 6 CPU treads per MPI Rank (e.g. for a GPU run with spare CPU resources):

  <diag_name>.adios2_operator.type = blosc
  <diag_name>.adios2_operator.parameters.compressor = zstd
  <diag_name>.adios2_operator.parameters.clevel = 1
  <diag_name>.adios2_operator.parameters.doshuffle = BLOSC_BITSHUFFLE
  <diag_name>.adios2_operator.parameters.threshold = 2048
  <diag_name>.adios2_operator.parameters.nthreads = 6  # per MPI rank (and thus per GPU)
  

  or for the lossy ZFP compressor using very strong compression per scalar:

  <diag_name>.adios2_operator.type = zfp
  <diag_name>.adios2_operator.parameters.precision = 3
  

• <diag_name>.adios2_engine.type (bp4, sst, ssc, dataman) optional,

  ADIOS2 Engine type for openPMD data dumps. See full list of engines at ADIOS2 readthedocs

• <diag_name>.adios2_engine.parameters.* optional,

  ADIOS2 Engine parameters for openPMD data dumps.

  An example for parameters for the BP engine are setting the number of writers (NumAggregators), transparently redirecting data to burst buffers etc. A detailed list of engine-specific parameters are available at the official ADIOS2 documentation

  <diag_name>.adios2_engine.parameters.NumAggregators = 2048
  <diag_name>.adios2_engine.parameters.BurstBufferPath="/mnt/bb/username"
  

• <diag_name>.fields_to_plot (list of strings, optional)

  Fields written to output. Possible scalar fields: part_per_cell rho phi F part_per_grid divE divB and rho_<species_name>, where <species_name> must match the name of one of the available particle species. Note that phi will only be written out when do_electrostatic==labframe. Also, note that for <diag_name>.diag_type = BackTransformed, the only scalar field currently supported is rho. Possible vector field components in Cartesian geometry: Ex Ey Ez Bx By Bz jx jy jz. Possible vector field components in RZ geometry: Er Et Ez Br Bt Bz jr jt jz. The default is <diag_name>.fields_to_plot = Ex Ey Ez Bx By Bz jx jy jz in Cartesian geometry and <diag_name>.fields_to_plot = Er Et Ez Br Bt Bz jr jt jz in RZ geometry. When the special value none is specified, no fields are written out. Note that the fields are averaged on the cell centers before they are written to file. Otherwise, we reconstruct a 2D Cartesian slice of the fields for output at \(\theta=0\).

• <diag_name>.dump_rz_modes (0 or 1) optional (default 0)

  Whether to save all modes when in RZ. When openpmd_backend = openpmd, this parameter is ignored and all modes are saved.

• <diag_name>.particle_fields_to_plot (list of strings, optional)

  Names of per-cell diagnostics of particle properties to calculate and output as additional fields. Note that the deposition onto the grid does not respect the particle shape factor, but instead uses nearest-grid point interpolation. Default is none. Parser functions for these field names are specified by <diag_name>.particle_fields.<field_name>(x,y,z,ux,uy,uz). Also, note that this option is only available for <diag_name>.diag_type = Full

• <diag_name>.particle_fields_species (list of strings, optional)

  Species for which to calculate particle_fields_to_plot. Fields will be calculated separately for each specified species. The default is a list of all of the available particle species.

• <diag_name>.particle_fields.<field_name>.do_average (0 or 1) optional (default 1)

  Whether the diagnostic is an average or a sum. With an average, the sum over the specified function is divided by the sum of the particle weights in each cell.

• <diag_name>.particle_fields.<field_name>(x,y,z,ux,uy,uz) (parser string)

  Parser function to be calculated for each particle per cell. The averaged field written is

  \[\texttt{<field_name>_<species>} = \frac{\sum_{i=1}^N w_i \, f(x_i,y_i,z_i,u_{x,i},u_{y,i},u_{z,i})}{\sum_{i=1}^N w_i}\]

  where \(w_i\) is the particle weight, \(f()\) is the parser function, and \((x_i,y_i,z_i)\) are particle positions in units of a meter. The sums are over all particles of type <species> in a cell (ignoring the particle shape factor) that satisfy <diag_name>.particle_fields.<field_name>.filter(x,y,z,ux,uy,uz). When <diag_name>.particle_fields.<field_name>.do_average is 0, the division by the sum over particle weights is not done. In 1D or 2D, the particle coordinates will follow the WarpX convention. \((u_{x,i},u_{y,i},u_{z,i})\) are components of the particle four-momentum. \(u = \gamma v/c\), \(\gamma\) is the Lorentz factor, \(v\) is the particle velocity and \(c\) is the speed of light. For photons, we use the standardized momentum \(u = p/(m_{e}c)\), where \(p\) is the momentum of the photon and \(m_{e}\) the mass of an electron.

• <diag_name>.particle_fields.<field_name>.filter(x,y,z,ux,uy,uz) (parser string, optional)

  Parser function returning a boolean for whether to include a particle in the diagnostic. If not specified, all particles will be included (see above). The function arguments are the same as above.

• <diag_name>.plot_raw_fields (0 or 1) optional (default 0)

  By default, the fields written in the plot files are averaged on the cell centers. When <diag_name>.plot_raw_fields = 1, then the raw (i.e. non-averaged) fields are also saved in the output files. Only works with <diag_name>.format = plotfile. See this section in the yt documentation for more details on how to view raw fields.

• <diag_name>.plot_raw_fields_guards (0 or 1) optional (default 0)

  Only used when <diag_name>.plot_raw_fields = 1. Whether to include the guard cells in the output of the raw fields. Only works with <diag_name>.format = plotfile.

• <diag_name>.coarsening_ratio (list of int) optional (default 1 1 1)

  Reduce size of the selected diagnostic fields output by this ratio in each dimension. (For a ratio of N, this is done by averaging the fields over N or (N+1) points depending on the staggering). If blocking_factor and max_grid_size are used for the domain decomposition, as detailed in the domain decomposition section, coarsening_ratio should be an integer divisor of blocking_factor. If warpx.numprocs is used instead, the total number of cells in a given dimension must be a multiple of the coarsening_ratio multiplied by numprocs in that dimension.

• <diag_name>.file_prefix (string) optional (default diags/<diag_name>)

  Root for output file names. Supports sub-directories.

• <diag_name>.file_min_digits (int) optional (default 6)

  The minimum number of digits used for the iteration number appended to the diagnostic file names.

• <diag_name>.diag_lo (list float, 1 per dimension) optional (default -infinity -infinity -infinity)

  Lower corner of the output fields (if smaller than warpx.dom_lo, then set to warpx.dom_lo). Currently, when the diag_lo is different from warpx.dom_lo, particle output is disabled.

• <diag_name>.diag_hi (list float, 1 per dimension) optional (default +infinity +infinity +infinity)

  Higher corner of the output fields (if larger than warpx.dom_hi, then set to warpx.dom_hi). Currently, when the diag_hi is different from warpx.dom_hi, particle output is disabled.

• <diag_name>.write_species (0 or 1) optional (default 1)

  Whether to write species output or not. For checkpoint format, always set this parameter to 1.

• <diag_name>.species (list of string, default all physical species in the simulation)

  Which species dumped in this diagnostics.

• <diag_name>.<species_name>.variables (list of strings separated by spaces, optional)

  List of particle quantities to write to output. Choices are w for the particle weight and ux uy uz for the particle momenta. When using the lab-frame electrostatic solver, phi (electrostatic potential, on the macroparticles) is also available. By default, all particle quantities (except phi) are written. If <diag_name>.<species_name>.variables = none, no particle data are written, except for particle positions, which are always included.

• <diag_name>.<species_name>.random_fraction (float) optional

  If provided <diag_name>.<species_name>.random_fraction = a, only a fraction of the particle data of this species will be dumped randomly in diag <diag_name>, i.e. if rand() < a, this particle will be dumped, where rand() denotes a random number generator. The value a provided should be between 0 and 1.

• <diag_name>.<species_name>.uniform_stride (int) optional

  If provided <diag_name>.<species_name>.uniform_stride = n, every n particle of this species will be dumped, selected uniformly. The value provided should be an integer greater than or equal to 0.

• <diag_name>.<species_name>.plot_filter_function(t,x,y,z,ux,uy,uz) (string) optional

  Users can provide an expression returning a boolean for whether a particle is dumped. t represents the physical time in seconds during the simulation. x, y, z represent particle positions in the unit of meter. ux, uy, uz represent particle momenta in the unit of \(\gamma v/c\), where \(\gamma\) is the Lorentz factor, \(v/c\) is the particle velocity normalized by the speed of light. E.g. If provided (x>0.0)*(uz<10.0) only those particles located at positions x greater than 0, and those having momentum uz less than 10, will be dumped.

• amrex.async_out (0 or 1) optional (default 0)

  Whether to use asynchronous IO when writing plotfiles. This only has an effect when using the AMReX plotfile format. Please see the data analysis section for more information.

• amrex.async_out_nfiles (int) optional (default 64)

  The maximum number of files to write to when using asynchronous IO. To use asynchronous IO with more than amrex.async_out_nfiles MPI ranks, WarpX must be configured with -DWarpX_MPI_THREAD_MULTIPLE=ON. Please see the data analysis section for more information.

• warpx.field_io_nfiles and warpx.particle_io_nfiles (int) optional (default 1024)

  The maximum number of files to use when writing field and particle data to plotfile directories.

• warpx.mffile_nstreams (int) optional (default 4)

  Limit the number of concurrent readers per file.

BackTransformed Diagnostics

BackTransformed diag type are used when running a simulation in a boosted frame, to reconstruct output data to the lab frame. This option can be set using <diag_name>.diag_type = BackTransformed. We support the following list of options from Full Diagnostics

<diag_name>.format, <diag_name>.openpmd_backend, <diag_name>.dump_rz_modes, <diag_name>.file_prefix, <diag_name>.diag_lo, <diag_name>.diag_hi, <diag_name>.write_species, <diag_name>.species.

Additional options for this diagnostic include:

• <diag_name>.num_snapshots_lab (integer)

  Only used when <diag_name>.diag_type is BackTransformed. The number of lab-frame snapshots that will be written. Only this option or intervals should be specified; a run-time error occurs if the user attempts to set both num_snapshots_lab and intervals.

• <diag_name>.intervals (string)

  Only used when <diag_name>.diag_type is BackTransformed. Using the Intervals parser syntax, this string defines the lab frame times at which data is dumped, given as multiples of the step size dt_snapshots_lab or dz_snapshots_lab described below. Example: btdiag1.intervals = 10:11,20:24:2 and btdiag1.dt_snapshots_lab = 1.e-12 indicate to dump at lab times 1e-11, 1.1e-11, 2e-11, 2.2e-11, and 2.4e-11 seconds. Note that the stop interval, the second number in the slice, must always be specified. Only this option or num_snapshots_lab should be specified; a run-time error occurs if the user attempts to set both num_snapshots_lab and intervals.

• <diag_name>.dt_snapshots_lab (float, in seconds)

  Only used when <diag_name>.diag_type is BackTransformed. The time interval in between the lab-frame snapshots (where this time interval is expressed in the laboratory frame).

• <diag_name>.dz_snapshots_lab (float, in meters)

  Only used when <diag_name>.diag_type is BackTransformed. Distance between the lab-frame snapshots (expressed in the laboratory frame). dt_snapshots_lab is then computed by dt_snapshots_lab = dz_snapshots_lab/c. Either dt_snapshots_lab or dz_snapshot_lab is required.

• <diag_name>.buffer_size (integer)

  Only used when <diag_name>.diag_type is BackTransformed. The default size of the back transformed diagnostic buffers used to generate lab-frame data is 256. That is, when the multifab with lab-frame data has 256 z-slices, the data will be flushed out. However, if many lab-frame snapshots are required for diagnostics and visualization, the GPU may run out of memory with many large boxes with a size of 256 in the z-direction. This input parameter can then be used to set a smaller buffer-size, preferably multiples of 8, such that, a large number of lab-frame snapshot data can be generated without running out of gpu memory. The downside to using a small buffer size, is that the I/O time may increase due to frequent flushes of the lab-frame data. The other option is to keep the default value for buffer size and use slices to reduce the memory footprint and maintain optimum I/O performance.

• <diag_name>.do_back_transformed_fields (0 or 1) optional (default 1)

  Only used when <diag_name>.diag_type is BackTransformed Whether to back transform the fields or not. Note that for BackTransformed diagnostics, at least one of the options <diag_name>.do_back_transformed_fields or <diag_name>.do_back_transformed_particles must be 1.

• <diag_name>.do_back_transformed_particles (0 or 1) optional (default 1)

  Only used when <diag_name>.diag_type is BackTransformed Whether to back transform the particle data or not. Note that for BackTransformed diagnostics, at least one of the options <diag_name>.do_back_transformed_fields or <diag_name>.do_back_transformed_particles must be 1. If diag_name.write_species = 0, then <diag_name>.do_back_transformed_particles will be set to 0 in the simulation and particles will not be backtransformed.

Boundary Scraping Diagnostics

BoundaryScrapingDiagnostics are used to collect the particles that are absorbed at the boundaries, throughout the simulation. This diagnostic type is specified by setting <diag_name>.diag_type = BoundaryScraping. Currently, the only supported output format is openPMD, so the user also needs to set <diag>.format=openpmd and WarpX must be compiled with openPMD turned on. The data that is to be collected and recorded is controlled per species and per boundary by setting one or more of the flags to 1, <species>.save_particles_at_xlo/ylo/zlo, <species>.save_particles_at_xhi/yhi/zhi, and <species>.save_particles_at_eb. (Note that this diagnostics does not save any field ; it only saves particles.)

The data collected at each boundary is written out to a subdirectory of the diagnostics directory with the name of the boundary, for example, particles_at_xlo, particles_at_zhi, or particles_at_eb. By default, all of the collected particle data is written out at the end of the simulation. Optionally, the <diag_name>.intervals parameter can be given to specify writing out the data more often. This can be important if a large number of particles are lost, avoiding filling up memory with the accumulated lost particle data.

In addition to their usual attributes, the saved particles have

an integer attribute stepScraped, which indicates the PIC iteration at which each particle was absorbed at the boundary, a real attribute deltaTimeScraped, which indicates the time between the time associated to stepScraped and the exact time when each particle hits the boundary. 3 real attributes nx, ny, nz, which represents the three components of the normal to the boundary on the point of contact of the particles (not saved if they reach non-EB boundaries)

BoundaryScrapingDiagnostics can be used with <diag_name>.<species>.random_fraction, <diag_name>.<species>.uniform_stride, and <diag_name>.<species>.plot_filter_function, which have the same behavior as for FullDiagnostics. For BoundaryScrapingDiagnostics, these filters are applied at the time the data is written to file. An implication of this is that more particles may initially be accumulated in memory than are ultimately written. t in plot_filter_function refers to the time the diagnostic is written rather than the time the particle crossed the boundary.

Reduced Diagnostics

ReducedDiags allow the user to compute some reduced quantity (particle temperature, max of a field) and write a small amount of data to text files.

• warpx.reduced_diags_names (strings, separated by spaces)

  The names given by the user of simple reduced diagnostics. Also the names of the output .txt files. This reduced diagnostics aims to produce simple outputs of the time history of some physical quantities. If warpx.reduced_diags_names is not provided in the input file, no reduced diagnostics will be done. This is then used in the rest of the input deck; in this documentation we use <reduced_diags_name> as a placeholder.

• <reduced_diags_name>.type (string)

  The type of reduced diagnostics associated with this <reduced_diags_name>. For example, ParticleEnergy, FieldEnergy, etc. All available types are described below in detail. For all reduced diagnostics, the first and the second columns in the output file are the time step and the corresponding physical time in seconds, respectively.

  • ParticleEnergy

    This type computes the total and mean relativistic particle kinetic energy among all species:

    \[E_p = \sum_{i=1}^N w_i \, \left( \sqrt{|\boldsymbol{p}_i|^2 c^2 + m_0^2 c^4} - m_0 c^2 \right)\]

    where \(\boldsymbol{p}_i\) is the relativistic momentum of the \(i\)-th particle, \(c\) is the speed of light, \(m_0\) is the rest mass, \(N\) is the number of particles, and \(w_i\) is the weight of the \(i\)-th particle.

    The output columns are the total energy of all species, the total energy per species, the total mean energy \(E_p / \sum_i w_i\) of all species, and the total mean energy per species.

  • ParticleMomentum

    This type computes the total and mean relativistic particle momentum among all species:

    \[\boldsymbol{P}_p = \sum_{i=1}^N w_i \, \boldsymbol{p}_i\]

    where \(\boldsymbol{p}_i\) is the relativistic momentum of the \(i\)-th particle, \(N\) is the number of particles, and \(w_i\) is the weight of the \(i\)-th particle.

    The output columns are the components of the total momentum of all species, the total momentum per species, the total mean momentum \(\boldsymbol{P}_p / \sum_i w_i\) of all species, and the total mean momentum per species.

  • FieldEnergy

    This type computes the electromagnetic field energy

    \[E_f = \frac{1}{2} \sum_{\text{cells}} \left( \varepsilon_0 |\boldsymbol{E}|^2 + \frac{|\boldsymbol{B}|^2}{\mu_0} \right) \Delta V\]

    where \(\boldsymbol{E}\) is the electric field, \(\boldsymbol{B}\) is the magnetic field, \(\varepsilon_0\) is the vacuum permittivity, \(\mu_0\) is the vacuum permeability, \(\Delta V\) is the cell volume (or cell area in 2D), and the sum is over all cells.

    The output columns are the total field energy \(E_f\), the \(\boldsymbol{E}\) field energy, and the \(\boldsymbol{B}\) field energy, at each mesh refinement level.

  • FieldMomentum

    This type computes the electromagnetic field momentum

    \[\boldsymbol{P}_f = \varepsilon_0 \sum_{\text{cells}} \left( \boldsymbol{E} \times \boldsymbol{B} \right) \Delta V\]

    where \(\boldsymbol{E}\) is the electric field, \(\boldsymbol{B}\) is the magnetic field, \(\varepsilon_0\) is the vacuum permittivity, \(\Delta V\) is the cell volume (or cell area in 2D), and the sum is over all cells.

    The output columns are the components of the total field momentum \(\boldsymbol{P}_f\) at each mesh refinement level.

    Note that the fields are not averaged on the cell centers before their energy is computed.

  • FieldMaximum

    This type computes the maximum value of each component of the electric and magnetic fields and of the norm of the electric and magnetic field vectors. Measuring maximum fields in a plasma might be very noisy in PIC, use this instead for analysis of scenarios such as an electromagnetic wave propagating in vacuum.

    The output columns are the maximum value of the \(E_x\) field, the maximum value of the \(E_y\) field, the maximum value of the \(E_z\) field, the maximum value of the norm \(|E|\) of the electric field, the maximum value of the \(B_x\) field, the maximum value of the \(B_y\) field, the maximum value of the \(B_z\) field and the maximum value of the norm \(|B|\) of the magnetic field, at mesh refinement levels from 0 to \(n\).

    Note that the fields are averaged on the cell centers before their maximum values are computed.

  • FieldProbe

    This type computes the value of each component of the electric and magnetic fields and of the Poynting vector (a measure of electromagnetic flux) at points in the domain.

    Multiple geometries for point probes can be specified via <reduced_diags_name>.probe_geometry = ...:

    • Point (default): a single point

    • Line: a line of points with equal spacing

    • Plane: a plane of points with equal spacing

    Point: The point where the fields are measured is specified through the input parameters <reduced_diags_name>.x_probe, <reduced_diags_name>.y_probe and <reduced_diags_name>.z_probe.

    Line: probe a 1 dimensional line of points to create a line detector. Initial input parameters x_probe, y_probe, and z_probe designate one end of the line detector, while the far end is specified via <reduced_diags_name>.x1_probe, <reduced_diags_name>.y1_probe, <reduced_diags_name>.z1_probe. Additionally, <reduced_diags_name>.resolution must be defined to give the number of detector points along the line (equally spaced) to probe.

    Plane: probe a 2 dimensional plane of points to create a square plane detector. Initial input parameters x_probe, y_probe, and z_probe designate the center of the detector. The detector plane is normal to a vector specified by <reduced_diags_name>.target_normal_x, <reduced_diags_name>.target_normal_y, and <reduced_diags_name>.target_normal_z. Note that it is not necessary to specify the target_normal vector in a 2D simulation (the only supported normal is in y). The top of the plane is perpendicular to an “up” vector denoted by <reduced_diags_name>.target_up_x, <reduced_diags_name>.target_up_y, and <reduced_diags_name>.target_up_z. The detector has a square radius to be determined by <reduced_diags_name>.detector_radius. Similarly to the line detector, the plane detector requires a resolution <reduced_diags_name>.resolution, which denotes the number of detector particles along each side of the square detector.

    The output columns are the value of the \(E_x\) field, the value of the \(E_y\) field, the value of the \(E_z\) field, the value of the \(B_x\) field, the value of the \(B_y\) field, the value of the \(B_z\) field and the value of the Poynting Vector \(|S|\) of the electromagnetic fields, at mesh refinement levels from 0 to \(n\), at point (\(x\), \(y\), \(z\)).

    The fields are always interpolated to the measurement point. The interpolation order can be set by specifying <reduced_diags_name>.interp_order, defaulting to 1. In RZ geometry, this only saves the 0’th azimuthal mode component of the fields. Time integrated electric and magnetic field components can instead be obtained by specifying <reduced_diags_name>.integrate = true. The integration is done every time step even when the data is written out less often. In a moving window simulation, the FieldProbe can be set to follow the moving frame by specifying <reduced_diags_name>.do_moving_window_FP = 1 (default 0).

    Warning

    The FieldProbe reduced diagnostic does not yet add a Lorentz back transformation for boosted frame simulations. Thus, it records field data in the boosted frame, not (yet) in the lab frame.

  • RhoMaximum

    This type computes the maximum and minimum values of the total charge density as well as the maximum absolute value of the charge density of each charged species. Please be aware that measuring maximum charge densities might be very noisy in PIC simulations.

    The output columns are the maximum value of the \(rho\) field, the minimum value of the \(rho\) field, the maximum value of the absolute \(|rho|\) field of each charged species.

    Note that the charge densities are averaged on the cell centers before their maximum values are computed.

  • FieldReduction

    This type computes an arbitrary reduction of the positions, the current density, and the electromagnetic fields.

    • <reduced_diags_name>.reduced_function(x,y,z,Ex,Ey,Ez,Bx,By,Bz,jx,jy,jz) (string)

      An analytic function to be reduced must be provided, using the math parser.

    • <reduced_diags_name>.reduction_type (string)

      The type of reduction to be performed. It must be either Maximum, Minimum or Integral. Integral computes the spatial integral of the function defined in the parser by summing its value on all grid points and multiplying the result by the volume of a cell. Please be also aware that measuring maximum quantities might be very noisy in PIC simulations.

    The only output column is the reduced value.

    Note that the fields are averaged on the cell centers before the reduction is performed.

  • ParticleNumber

    This type computes the total number of macroparticles and of physical particles (i.e. the sum of their weights) in the whole simulation domain (for each species and summed over all species). It can be useful in particular for simulations with creation (ionization, QED processes) or removal (resampling) of particles.

    The output columns are total number of macroparticles summed over all species, total number of macroparticles of each species, sum of the particles’ weight summed over all species, sum of the particles’ weight of each species.

  • BeamRelevant

    This type computes properties of a particle beam relevant for particle accelerators, like position, momentum, emittance, etc.

    <reduced_diags_name>.species must be provided, such that the diagnostics are done for this (beam-like) species only.

    The output columns (for 3D-XYZ) are the following, where the average is done over the whole species (typical usage: the particle beam is in a separate species):

    [0]: simulation step (iteration).

    [1]: time (s).

    [2], [3], [4]: The mean values of beam positions (m) \(\langle x \rangle\), \(\langle y \rangle\), \(\langle z \rangle\).

    [5], [6], [7]: The mean values of beam relativistic momenta (kg m/s) \(\langle p_x \rangle\), \(\langle p_y \rangle\), \(\langle p_z \rangle\).

    [8]: The mean Lorentz factor \(\langle \gamma \rangle\).

    [9], [10], [11]: The RMS values of beam positions (m) \(\delta_x = \sqrt{ \langle (x - \langle x \rangle)^2 \rangle }\), \(\delta_y = \sqrt{ \langle (y - \langle y \rangle)^2 \rangle }\), \(\delta_z = \sqrt{ \langle (z - \langle z \rangle)^2 \rangle }\).

    [12], [13], [14]: The RMS values of beam relativistic momenta (kg m/s) \(\delta_{px} = \sqrt{ \langle (p_x - \langle p_x \rangle)^2 \rangle }\), \(\delta_{py} = \sqrt{ \langle (p_y - \langle p_y \rangle)^2 \rangle }\), \(\delta_{pz} = \sqrt{ \langle (p_z - \langle p_z \rangle)^2 \rangle }\).

    [15]: The RMS value of the Lorentz factor \(\sqrt{ \langle (\gamma - \langle \gamma \rangle)^2 \rangle }\).

    [16], [17], [18]: beam projected transverse RMS normalized emittance (m) \(\epsilon_x = \dfrac{1}{mc} \sqrt{\delta_x^2 \delta_{px}^2 - \Big\langle (x-\langle x \rangle) (p_x-\langle p_x \rangle) \Big\rangle^2}\), \(\epsilon_y = \dfrac{1}{mc} \sqrt{\delta_y^2 \delta_{py}^2 - \Big\langle (y-\langle y \rangle) (p_y-\langle p_y \rangle) \Big\rangle^2}\), \(\epsilon_z = \dfrac{1}{mc} \sqrt{\delta_z^2 \delta_{pz}^2 - \Big\langle (z-\langle z \rangle) (p_z-\langle p_z \rangle) \Big\rangle^2}\).

    [19], [20]: Twiss alpha for the transverse directions \(\alpha_x = - \Big\langle (x-\langle x \rangle) (p_x-\langle p_x \rangle) \Big\rangle \Big/ \epsilon_x\), \(\alpha_y = - \Big\langle (y-\langle y \rangle) (p_y-\langle p_y \rangle) \Big\rangle \Big/ \epsilon_y\).

    [21], [22]: beta function for the transverse directions (m) \(\beta_x = \dfrac{{\delta_x}^2}{\epsilon_x}\), \(\beta_y = \dfrac{{\delta_y}^2}{\epsilon_y}\).

    [23]: The charge of the beam (C).

    For 2D-XZ, \(\langle y \rangle\), \(\delta_y\), and \(\epsilon_y\) will not be outputted.

  • LoadBalanceCosts

    This type computes the cost, used in load balancing, for each box on the domain. The cost \(c\) is computed as

    \[c = n_{\text{particle}} \cdot w_{\text{particle}} + n_{\text{cell}} \cdot w_{\text{cell}},\]

    where \(n_{\text{particle}}\) is the number of particles on the box, \(w_{\text{particle}}\) is the particle cost weight factor (controlled by algo.costs_heuristic_particles_wt), \(n_{\text{cell}}\) is the number of cells on the box, and \(w_{\text{cell}}\) is the cell cost weight factor (controlled by algo.costs_heuristic_cells_wt).

  • LoadBalanceEfficiency

    This type computes the load balance efficiency, given the present costs and distribution mapping. Load balance efficiency is computed as the mean cost over all ranks, divided by the maximum cost over all ranks. Until costs are recorded, load balance efficiency is output as -1; at earliest, the load balance efficiency can be output starting at step 2, since costs are not recorded until step 1.

  • ParticleHistogram

    This type computes a user defined particle histogram.

    • <reduced_diags_name>.species (string)

      A species name must be provided, such that the diagnostics are done for this species.

    • <reduced_diags_name>.histogram_function(t,x,y,z,ux,uy,uz) (string)

      A histogram function must be provided. t represents the physical time in seconds during the simulation. x, y, z represent particle positions in the unit of meter. ux, uy, uz represent the particle momenta in the unit of \(\gamma v/c\), where \(\gamma\) is the Lorentz factor, \(v/c\) is the particle velocity normalized by the speed of light. E.g. x produces the position (density) distribution in x. ux produces the momentum distribution in x, sqrt(ux*ux+uy*uy+uz*uz) produces the speed distribution. The default value of the histogram without normalization is \(f = \sum\limits_{i=1}^N w_i\), where \(\sum\limits_{i=1}^N\) is the sum over \(N\) particles in that bin, \(w_i\) denotes the weight of the ith particle.

    • <reduced_diags_name>.bin_number (int > 0)

      This is the number of bins used for the histogram.

    • <reduced_diags_name>.bin_max (float)

      This is the maximum value of the bins.

    • <reduced_diags_name>.bin_min (float)

      This is the minimum value of the bins.

    • <reduced_diags_name>.normalization (optional)

      This provides options to normalize the histogram:

      unity_particle_weight uses unity particle weight to compute the histogram, such that the values of the histogram are the number of counted macroparticles in that bin, i.e. \(f = \sum\limits_{i=1}^N 1\), \(N\) is the number of particles in that bin.

      max_to_unity will normalize the histogram such that its maximum value is one.

      area_to_unity will normalize the histogram such that the area under the histogram is one, so the histogram is also the probability density function.

      If nothing is provided, the macroparticle weight will be used to compute the histogram, and no normalization will be done.

    • <reduced_diags_name>.filter_function(t,x,y,z,ux,uy,uz) (string) optional

      Users can provide an expression returning a boolean for whether a particle is taken into account when calculating the histogram. t represents the physical time in seconds during the simulation. x, y, z represent particle positions in the unit of meter. ux, uy, uz represent particle momenta in the unit of \(\gamma v/c\), where \(\gamma\) is the Lorentz factor, \(v/c\) is the particle velocity normalized by the speed of light. E.g. If provided (x>0.0)*(uz<10.0) only those particles located at positions x greater than 0, and those having momentum uz less than 10, will be taken into account when calculating the histogram.

    The output columns are values of the 1st bin, the 2nd bin, …, the nth bin. An example input file and a loading python script of using the histogram reduced diagnostics are given in Examples/Tests/initial_distribution/.

  • ParticleHistogram2D

    This type computes a user defined, 2D particle histogram.

    • <reduced_diags_name>.species (string)

      A species name must be provided, such that the diagnostics are done for this species.

    • <reduced_diags_name>.file_min_digits (int) optional (default 6)

      The minimum number of digits used for the iteration number appended to the diagnostic file names.

    • <reduced_diags_name>.histogram_function_abs(t,x,y,z,ux,uy,uz,w) (string)

      A histogram function must be provided for the abscissa axis. t represents the physical time in seconds during the simulation. x, y, z represent particle positions in the unit of meter. ux, uy, uz represent the particle velocities in the unit of \(\gamma v/c\), where \(\gamma\) is the Lorentz factor, \(v/c\) is the particle velocity normalized by the speed of light. w represents the weight.

    • <reduced_diags_name>.histogram_function_ord(t,x,y,z,ux,uy,uz,w) (string)

      A histogram function must be provided for the ordinate axis.

    • <reduced_diags_name>.bin_number_abs (int > 0) and <reduced_diags_name>.bin_number_ord (int > 0)

      These are the number of bins used for the histogram for the abscissa and ordinate axis respectively.

    • <reduced_diags_name>.bin_max_abs (float) and <reduced_diags_name>.bin_max_ord (float)

      These are the maximum value of the bins for the abscissa and ordinate axis respectively. Particles with values outside of these ranges are discarded.

    • <reduced_diags_name>.bin_min_abs (float) and <reduced_diags_name>.bin_min_ord (float)

      These are the minimum value of the bins for the abscissa and ordinate axis respectively. Particles with values outside of these ranges are discarded.

    • <reduced_diags_name>.filter_function(t,x,y,z,ux,uy,uz,w) (string) optional

      Users can provide an expression returning a boolean for whether a particle is taken into account when calculating the histogram. t represents the physical time in seconds during the simulation. x, y, z represent particle positions in the unit of meter. ux, uy, uz represent particle velocities in the unit of \(\gamma v/c\), where \(\gamma\) is the Lorentz factor, \(v/c\) is the particle velocity normalized by the speed of light. w represents the weight.

    • <reduced_diags_name>.value_function(t,x,y,z,ux,uy,uz,w) (string) optional

      Users can provide an expression for the weight used to calculate the number of particles per cell associated with the selected abscissa and ordinate functions and/or the filter function. t represents the physical time in seconds during the simulation. x, y, z represent particle positions in the unit of meter. ux, uy, uz represent particle velocities in the unit of \(\gamma v/c\), where \(\gamma\) is the Lorentz factor, \(v/c\) is the particle velocity normalized by the speed of light. w represents the weight.

    The output is a <reduced_diags_name> folder containing a set of openPMD files. An example input file and a loading python script of using the histogram2D reduced diagnostics are given in Examples/Tests/histogram2D/.

  • ParticleExtrema

    This type computes the minimum and maximum values of particle position, momentum, gamma, weight, and the \(\chi\) parameter for QED species.

    <reduced_diags_name>.species must be provided, such that the diagnostics are done for this species only.

    The output columns are minimum and maximum position \(x\), \(y\), \(z\); minimum and maximum momentum \(p_x\), \(p_y\), \(p_z\); minimum and maximum gamma \(\gamma\); minimum and maximum weight \(w\); minimum and maximum \(\chi\).

    Note that when the QED parameter \(\chi\) is computed, field gather is carried out at every output, so the time of the diagnostic may be long depending on the simulation size.

  • ChargeOnEB

    This type computes the total surface charge on the embedded boundary (in Coulombs), by using the formula

    \[Q_{tot} = \epsilon_0 \iint dS \cdot E\]

    where the integral is performed over the surface of the embedded boundary.

    When providing <reduced_diags_name>.weighting_function(x,y,z), the computed integral is weighted:

    \[Q = \epsilon_0 \iint dS \cdot E \times weighting(x, y, z)\]

    In particular, by choosing a weighting function which returns either 1 or 0, it is possible to compute the charge on only some part of the embedded boundary.

  • ColliderRelevant

    This diagnostics computes properties of two colliding beams that are relevant for particle colliders. Two species must be specified. Photon species are not supported yet. It is assumed that the two species propagate and collide along the z direction. The output columns (for 3D-XYZ) are the following, where the minimum, average and maximum are done over the whole species:

    [0]: simulation step (iteration).

    [1]: time (s).

    [2]: time derivative of the luminosity (\(m^{-2}s^{-1}\)) defined as:

    \[\frac{dL}{dt} = 2 c \iiint n_1(x,y,z) n_2(x,y,z) dx dy dz\]

    where \(n_1\), \(n_2\) are the number densities of the two colliding species.

    [3], [4], [5]: If, QED is enabled, the minimum, average and maximum values of the quantum parameter \(\chi\) of species 1: \(\chi_{min}\), \(\langle \chi \rangle\), \(\chi_{max}\). If QED is not enabled, these numbers are not computed.

    [6], [7]: The average and standard deviation of the values of the transverse coordinate \(x\) (m) of species 1: \(\langle x \rangle\), \(\sqrt{\langle x- \langle x \rangle \rangle^2}\).

    [8], [9]: The average and standard deviation of the values of the transverse coordinate \(y\) (m) of species 1: \(\langle y \rangle\), \(\sqrt{\langle y- \langle y \rangle \rangle^2}\).

    [10], [11], [12], [13]: The minimum, average, maximum and standard deviation of the angle \(\theta_x = \angle (u_x, u_z)\) (rad) of species 1: \({\theta_x}_{min}\), \(\langle \theta_x \rangle\), \({\theta_x}_{max}\), \(\sqrt{\langle \theta_x- \langle \theta_x \rangle \rangle^2}\).

    [14], [15], [16], [17]: The minimum, average, maximum and standard deviation of the angle \(\theta_y = \angle (u_y, u_z)\) (rad) of species 1: \({\theta_y}_{min}\), \(\langle \theta_y \rangle\), \({\theta_y}_{max}\), \(\sqrt{\langle \theta_y- \langle \theta_y \rangle \rangle^2}\).

    [18], …, [32]: Analogous quantities for species 2.

    For 2D-XZ, \(y\)-related quantities are not outputted. For 1D-Z, \(x\)-related and \(y\)-related quantities are not outputted. RZ geometry is not supported yet.

• <reduced_diags_name>.intervals (string)

  Using the Intervals Parser syntax, this string defines the timesteps at which reduced diagnostics are written to file.

• <reduced_diags_name>.path (string) optional (default ./diags/reducedfiles/)

  The path that the output file will be stored.

• <reduced_diags_name>.extension (string) optional (default txt)

  The extension of the output file.

• <reduced_diags_name>.separator (string) optional (default a whitespace)

  The separator between row values in the output file. The default separator is a whitespace.

• <reduced_diags_name>.precision (integer) optional (default 14)

  The precision used when writing out the data to the text files.

Lookup tables and other settings for QED modules

Lookup tables store pre-computed values for functions used by the QED modules. This feature requires to compile with QED=TRUE (and also with QED_TABLE_GEN=TRUE for table generation)

• qed_bw.lookup_table_mode (string)

  There are three options to prepare the lookup table required by the Breit-Wheeler module:

  • builtin: a built-in table is used (Warning: the table gives reasonable results but its resolution is quite low).

  • generate: a new table is generated. This option requires Boost math library (version >= 1.66) and to compile with QED_TABLE_GEN=TRUE. All the following parameters must be specified (table 1 is used to evolve the optical depth of the photons, while table 2 is used for pair generation):

    • qed_bw.tab_dndt_chi_min (float): minimum chi parameter for lookup table 1 ( used for the evolution of the optical depth of the photons)

    • qed_bw.tab_dndt_chi_max (float): maximum chi parameter for lookup table 1

    • qed_bw.tab_dndt_how_many (int): number of points to be used for lookup table 1

    • qed_bw.tab_pair_chi_min (float): minimum chi parameter for lookup table 2 ( used for pair generation)

    • qed_bw.tab_pair_chi_max (float): maximum chi parameter for lookup table 2

    • qed_bw.tab_pair_chi_how_many (int): number of points to be used for chi axis in lookup table 2

    • qed_bw.tab_pair_frac_how_many (int): number of points to be used for the second axis in lookup table 2 (the second axis is the ratio between the quantum parameter of the less energetic particle of the pair and the quantum parameter of the photon).

    • qed_bw.save_table_in (string): where to save the lookup table

    Alternatively, the lookup table can be generated using a standalone tool (see qed tools section).

  • load: a lookup table is loaded from a pre-generated binary file. The following parameter must be specified:

    • qed_bw.load_table_from (string): name of the lookup table file to read from.

• qed_qs.lookup_table_mode (string)

  There are three options to prepare the lookup table required by the Quantum Synchrotron module:

  • builtin: a built-in table is used (Warning: the table gives reasonable results but its resolution is quite low).

  • generate: a new table is generated. This option requires Boost math library (version >= 1.66) and to compile with QED_TABLE_GEN=TRUE. All the following parameters must be specified (table 1 is used to evolve the optical depth of the particles, while table 2 is used for photon emission):

    • qed_qs.tab_dndt_chi_min (float): minimum chi parameter for lookup table 1 ( used for the evolution of the optical depth of electrons and positrons)

    • qed_qs.tab_dndt_chi_max (float): maximum chi parameter for lookup table 1

    • qed_qs.tab_dndt_how_many (int): number of points to be used for lookup table 1

    • qed_qs.tab_em_chi_min (float): minimum chi parameter for lookup table 2 ( used for photon emission)

    • qed_qs.tab_em_chi_max (float): maximum chi parameter for lookup table 2

    • qed_qs.tab_em_chi_how_many (int): number of points to be used for chi axis in lookup table 2

    • qed_qs.tab_em_frac_how_many (int): number of points to be used for the second axis in lookup table 2 (the second axis is the ratio between the quantum parameter of the photon and the quantum parameter of the charged particle).

    • qed_qs.tab_em_frac_min (float): minimum value to be considered for the second axis of lookup table 2

    • qed_qs.save_table_in (string): where to save the lookup table

    Alternatively, the lookup table can be generated using a standalone tool (see qed tools section).

  • load: a lookup table is loaded from a pre-generated binary file. The following parameter must be specified:

    • qed_qs.load_table_from (string): name of the lookup table file to read from.

• qed_bw.chi_min (float): minimum chi parameter to be considered by the Breit-Wheeler engine

  (suggested value : 0.01)

• qed_qs.chi_min (float): minimum chi parameter to be considered by the Quantum Synchrotron engine

  (suggested value : 0.001)

• qed_qs.photon_creation_energy_threshold (float) optional (default 2)

  Energy threshold for photon particle creation in *me*c^2 units.

• warpx.do_qed_schwinger (bool) optional (default 0)

  If this is 1, Schwinger electron-positron pairs can be generated in vacuum in the cells where the EM field is high enough. Activating the Schwinger process requires the code to be compiled with QED=TRUE and PICSAR. If warpx.do_qed_schwinger = 1, Schwinger product species must be specified with qed_schwinger.ele_product_species and qed_schwinger.pos_product_species. Schwinger process requires either warpx.grid_type = collocated or algo.field_gathering=momentum-conserving (so that different field components are computed at the same location in the grid) and does not currently support mesh refinement, cylindrical coordinates or single precision.

• qed_schwinger.ele_product_species (string)

  If Schwinger process is activated, an electron product species must be specified (the name of an existing electron species must be provided).

• qed_schwinger.pos_product_species (string)

  If Schwinger process is activated, a positron product species must be specified (the name of an existing positron species must be provided).

• qed_schwinger.y_size (float; in meters)

  If Schwinger process is activated with DIM=2D, a transverse size must be specified. It is used to convert the pair production rate per unit volume into an actual number of created particles. This value should correspond to the typical transverse extent for which the EM field has a very high value (e.g. the beam waist for a focused laser beam).

• qed_schwinger.xmin,ymin,zmin and qed_schwinger.xmax,ymax,zmax (float) optional (default unlimited)

  When qed_schwinger.xmin and qed_schwinger.xmax are set, they delimit the region within which Schwinger pairs can be created. The same is applicable in the other directions.

• qed_schwinger.threshold_poisson_gaussian (integer) optional (default 25)

  If the expected number of physical pairs created in a cell at a given timestep is smaller than this threshold, a Poisson distribution is used to draw the actual number of physical pairs created. Otherwise a Gaussian distribution is used. Note that, regardless of this parameter, the number of macroparticles created is at most one per cell per timestep per species (with a weight corresponding to the number of physical pairs created).

Checkpoints and restart

WarpX supports checkpoints/restart via AMReX. The checkpoint capability can be turned with regular diagnostics: <diag_name>.format = checkpoint.

• amr.restart (string)

  Name of the checkpoint file to restart from. Returns an error if the folder does not exist or if it is not properly formatted.

• warpx.write_diagnostics_on_restart (bool) optional (default false)

  When true, write the diagnostics after restart at the time of the restart.

Intervals parser

WarpX can parse time step interval expressions of the form start:stop:period, e.g. 1:2:3, 4::, 5:6, :, ::10. A comma is used as a separator between groups of intervals, which we call slices. The resulting time steps are the union set of all given slices. White spaces are ignored. A single slice can have 0, 1 or 2 colons :, just as numpy slices, but with inclusive upper bound for stop.

• For 0 colon the given value is the period

• For 1 colon the given string is of the type start:stop

• For 2 colons the given string is of the type start:stop:period

Any value that is not given is set to default. Default is 0 for the start, std::numeric_limits<int>::max() for the stop and 1 for the period. For the 1 and 2 colon syntax, actually having values in the string is optional (this means that ::5, 100 ::10 and 100 : are all valid syntaxes).

All values can be expressions that will be parsed in the same way as other integer input parameters.

Examples

• something_intervals = 50 -> do something at timesteps 0, 50, 100, 150, etc. (equivalent to something_intervals = ::50)

• something_intervals = 300:600:100 -> do something at timesteps 300, 400, 500 and 600.

• something_intervals = 300::50 -> do something at timesteps 300, 350, 400, 450, etc.

• something_intervals = 105:108,205:208 -> do something at timesteps 105, 106, 107, 108, 205, 206, 207 and 208. (equivalent to something_intervals = 105 : 108 : , 205 : 208 :)

• something_intervals = : or something_intervals = :: -> do something at every timestep.

• something_intervals = 167:167,253:253,275:425:50 do something at timesteps 167, 253, 275, 325, 375 and 425.

This is essentially the python slicing syntax except that the stop is inclusive (0:100 contains 100) and that no colon means that the given value is the period.

Note that if a given period is zero or negative, the corresponding slice is disregarded. For example, something_intervals = -1 deactivates something and something_intervals = ::-1,100:1000:25 is equivalent to something_intervals = 100:1000:25.

Testing and Debugging

When developing, testing and debugging WarpX, the following options can be considered.

• warpx.verbose (0 or 1; default is 1 for true)

  Controls how much information is printed to the terminal, when running WarpX.

• warpx.always_warn_immediately (0 or 1; default is 0 for false)

  If set to 1, WarpX immediately prints every warning message as soon as it is generated. It is mainly intended for debug purposes, in case a simulation crashes before a global warning report can be printed.

• warpx.abort_on_warning_threshold (string: low, medium or high) optional

  Optional threshold to abort as soon as a warning is raised. If the threshold is set, warning messages with priority greater than or equal to the threshold trigger an immediate abort. It is mainly intended for debug purposes, and is best used with warpx.always_warn_immediately=1.

• amrex.abort_on_unused_inputs (0 or 1; default is 0 for false)

  When set to 1, this option causes simulation to fail after its completion if there were unused parameters. It is mainly intended for continuous integration and automated testing to check that all tests and inputs are adapted to API changes.

• amrex.use_profiler_syncs (0 or 1; default is 0 for false)

  Adds a synchronization at the start of communication, so any load balance will be caught there (the timer is called SyncBeforeComms), then the comm operation will run. This will slow down the run.

• warpx.serialize_initial_conditions (0 or 1) optional (default 0)

  Serialize the initial conditions for reproducible testing, e.g, in our continuous integration tests. Mainly whether or not to use OpenMP threading for particle initialization.

• warpx.safe_guard_cells (0 or 1) optional (default 0)

  Run in safe mode, exchanging more guard cells, and more often in the PIC loop (for debugging).

• ablastr.fillboundary_always_sync (0 or 1) optional (default 0)

  Run all FillBoundary operations on MultiFab to force-synchronize shared nodal points. This slightly increases communication cost and can help to spot missing nodal_sync flags in these operations.

[1] (1,2)

J.-L. Vay. Simulation Of Beams Or Plasmas Crossing At Relativistic Velocity. Physics of Plasmas, 15(5):56701, May 2008. doi:10.1063/1.2837054.

[2]

J.P. Verboncoeur. Symmetric Spline Weighting for Charge and Current Density in Particle Simulation. Journal of Computational Physics, 174(1):421–427, 2001. doi:10.1006/jcph.2001.6923.

[3]

H. Wiedemann. Particle Accelerator Physics. Springer Cham, 2015. ISBN 978-3-319-18317-6. doi:10.1007/978-3-319-18317-6.

[4]

A. Muraviev, A. Bashinov, E. Efimenko, V. Volokitin, I. Meyerov, and A. Gonoskov. Strategies for particle resampling in PIC simulations. Computer Physics Communications, 262:107826, 2021. doi:10.1016/j.cpc.2021.107826.

[5] (1,2,3,4)

S. Akturk, X. Gu, E. Zeek, and R. Trebino. Pulse-front tilt caused by spatial and temporal chirp. Opt. Express, 12(19):4399–4410, Sep 2004. doi:10.1364/OPEX.12.004399.

[6] (1,2)

F. Pérez, L. Gremillet, A. Decoster, M. Drouin, and E. Lefebvre. Improved modeling of relativistic collisions and collisional ionization in particle-in-cell codes. Physics of Plasmas, 19(8):083104, Aug 2012. doi:10.1063/1.4742167.

[7] (1,2)

D. P. Higginson, A. Link, and A. Schmidt. A pairwise nuclear fusion algorithm for weighted particle-in-cell plasma simulations. Journal of Computational Physics, 388:439–453, Jul 2019. doi:10.1016/j.jcp.2019.03.020.

[8]

T. Z. Esirkepov. Exact Charge Conservation Scheme For Particle-In-Cell Simulation With An Arbitrary Form-Factor. Computer Physics Communications, 135(2):144–153, Apr 2001.

[9] (1,2,3,4)

J.-L. Vay, I. Haber, and B. B. Godfrey. A domain decomposition method for pseudo-spectral electromagnetic simulations of plasmas. Journal of Computational Physics, 243:260–268, Jun 2013. doi:10.1016/j.jcp.2013.03.010.

[10]

A. V. Higuera and J. R. Cary. Structure-preserving second-order integration of relativistic charged particle trajectories in electromagnetic fields. Physics of Plasmas, 24(5):052104, 04 2017. URL: https://doi.org/10.1063/1.4979989, arXiv:https://pubs.aip.org/aip/pop/article-pdf/doi/10.1063/1.4979989/15988441/052104\_1\_online.pdf, doi:10.1063/1.4979989.

[11]

B. M. Cowan, D. L. Bruhwiler, J. R. Cary, E. Cormier-Michel, and C. G. R. Geddes. Generalized algorithm for control of numerical dispersion in explicit time-domain electromagnetic simulations. Physical Review Special Topics-Accelerators And Beams, Apr 2013. doi:10.1103/PhysRevSTAB.16.041303.

[12]

T. Xiao and Q. H. Liu. An enlarged cell technique for the conformal FDTD method to model perfectly conducting objects. In 2005 IEEE Antennas and Propagation Society International Symposium, volume 1A, 122–125 Vol. 1A. 2005. doi:10.1109/APS.2005.1551259.

[13] (1,2,3)

R. Lehe, M. Kirchen, B. B. Godfrey, A. R. Maier, and J.-L. Vay. Elimination of numerical Cherenkov instability in flowing-plasma particle-in-cell simulations by using Galilean coordinates. Phys. Rev. E, 94:053305, Nov 2016. URL: https://link.aps.org/doi/10.1103/PhysRevE.94.053305, doi:10.1103/PhysRevE.94.053305.

[14]

A. Stanier, L. Chacón, and A. Le. A cancellation problem in hybrid particle-in-cell schemes due to finite particle size. Journal of Computational Physics, 420:109705, 2020. URL: https://www.sciencedirect.com/science/article/pii/S0021999120304794, doi:https://doi.org/10.1016/j.jcp.2020.109705.

[15]

B. B. Godfrey and J.-L. Vay. Numerical stability of relativistic beam multidimensional PIC simulations employing the Esirkepov algorithm. Journal of Computational Physics, 248:33–46, 2013.

[16]

T. Grismayer, R. Torres, P. Carneiro, F. Cruz, R. A. Fonseca, and L. O. Silva. Quantum Electrodynamics vacuum polarization solver. New Journal of Physics, 23(9):095005, Sep 2021. doi:10.1088/1367-2630/ac2004.