# Particle-in-Cell Method¶

In the *electromagnetic particle-in-cell method* (Birdsall and Langdon 1991),
the electromagnetic fields are solved on a grid, usually using Maxwell’s
equations

given here in natural units (\(\epsilon_0=\mu_0=c=1\)), where \(t\) is time, \(\mathbf{E}\) and \(\mathbf{B}\) are the electric and magnetic field components, and \(\rho\) and \(\mathbf{J}\) are the charge and current densities. The charged particles are advanced in time using the Newton-Lorentz equations of motion

where \(m\), \(q\), \(\mathbf{x}\), \(\mathbf{v}\) and \(\gamma=1/\sqrt{1-v^{2}}\) are respectively the mass, charge, position, velocity and relativistic factor of the particle given in natural units (\(c=1\)). The charge and current densities are interpolated on the grid from the particles’ positions and velocities, while the electric and magnetic field components are interpolated from the grid to the particles’ positions for the velocity update.

## Particle push¶

A centered finite-difference discretization of the Newton-Lorentz equations of motion is given by

In order to close the system, \(\bar{\mathbf{v}}^{i}\) must be expressed as a function of the other quantities. The two implementations that have become the most popular are presented below.

### Boris relativistic velocity rotation¶

The solution proposed by Boris (Boris 1970) is given by

where \(\bar{\gamma}^{i}\) is defined by \(\bar{\gamma}^{i} \equiv (\gamma^{i+1/2}+\gamma^{i-1/2} )/2\).

The system ([Eq:leapfrog_v],[Eq:boris_v]) is solved very efficiently following Boris’ method, where the electric field push is decoupled from the magnetic push. Setting \(\mathbf{u}=\gamma\mathbf{v}\), the velocity is updated using the following sequence:

where \(\mathbf{t}=\left(q\Delta t/2m\right)\mathbf{B}^{i}/\bar{\gamma}^{i}\) and where \(\bar{\gamma}^{i}\) can be calculated as \(\bar{\gamma}^{i}=\sqrt{1+(\mathbf{u}^-/c)^2}\).

The Boris implementation is second-order accurate, time-reversible and fast. Its implementation is very widespread and used in the vast majority of PIC codes.

### Vay Lorentz-invariant formulation¶

It was shown in (Vay 2008) that the Boris formulation is not Lorentz invariant and can lead to significant errors in the treatment of relativistic dynamics. A Lorentz invariant formulation is obtained by considering the following velocity average

This gives a system that is solvable analytically (see (Vay 2008) for a detailed derivation), giving the following velocity update:

where \(\mathbf{t}=\boldsymbol{\tau}/\gamma^{i+1/2}\), \(\boldsymbol{\tau}=\left(q\Delta t/2m\right)\mathbf{B}^{i}\), \(\gamma^{i+1/2}=\sqrt{\sigma+\sqrt{\sigma^{2}+\left(\tau^{2}+w^{2}\right)}}\), \(w=\mathbf{u^{*}}\cdot\boldsymbol{\tau}\), \(\sigma=\left(\gamma'^{2}-\tau^{2}\right)/2\) and \(\gamma'=\sqrt{1+(\mathbf{u}^{*}/c)^{2}}\). This Lorentz invariant formulation is particularly well suited for the modeling of ultra-relativistic charged particle beams, where the accurate account of the cancellation of the self-generated electric and magnetic fields is essential, as shown in (Vay 2008).

## Field solve¶

Various methods are available for solving Maxwell’s equations on a grid, based on finite-differences, finite-volume, finite-element, spectral, or other discretization techniques that apply most commonly on single structured or unstructured meshes and less commonly on multiblock multiresolution grid structures. In this chapter, we summarize the widespread second order finite-difference time-domain (FDTD) algorithm, its extension to non-standard finite-differences as well as the pseudo-spectral analytical time-domain (PSATD) and pseudo-spectral time-domain (PSTD) algorithms. Extension to multiresolution (or mesh refinement) PIC is described in, e.g. (Vay et al. 2012; Vay, Adam, and Heron 2004).

### Finite-Difference Time-Domain (FDTD)¶

The most popular algorithm for electromagnetic PIC codes is the Finite-Difference Time-Domain (or FDTD) solver

The differential operator is defined as \(\nabla=D_{x}\mathbf{\hat{x}}+D_{y}\mathbf{\hat{y}}+D_{z}\mathbf{\hat{z}}\) and the finite-difference operators in time and space are defined respectively as

and \(D_{x}G|_{i,j,k}^{n}=\left(G|_{i+1/2,j,k}^{n}-G|_{i-1/2,j,k}^{n}\right)/\Delta x\), where \(\Delta t\) and \(\Delta x\) are respectively the time step and the grid cell size along \(x\), \(n\) is the time index and \(i\), \(j\) and \(k\) are the spatial indices along \(x\), \(y\) and \(z\) respectively. The difference operators along \(y\) and \(z\) are obtained by circular permutation. The equations in brackets are given for completeness, as they are often not actually solved, thanks to the usage of a so-called charge conserving algorithm, as explained below. As shown in Figure [fig:yee_grid], the quantities are given on a staggered (or “Yee”) grid (Yee 1966), where the electric field components are located between nodes and the magnetic field components are located in the center of the cell faces. Knowing the current densities at half-integer steps, the electric field components are updated alternately with the magnetic field components at integer and half-integer steps respectively.

### Non-Standard Finite-Difference Time-Domain (NSFDTD)¶

In (Cole 1997, 2002), Cole introduced an implementation
of the source-free Maxwell’s wave equations for narrow-band applications
based on non-standard finite-differences (NSFD). In (Karkkainen et al. 2006),
Karkkainen *et al.* adapted it for wideband applications. At
the Courant limit for the time step and for a given set of parameters,
the stencil proposed in (Karkkainen et al. 2006) has no numerical dispersion
along the principal axes, provided that the cell size is the same
along each dimension (i.e. cubic cells in 3D). The “Cole-Karkkainnen”
(or CK) solver uses the non-standard finite difference formulation
(based on extended stencils) of the Maxwell-Ampere equation and can be
implemented as follows (Vay et al. 2011):

Eq. [Eq:Gauss] and [Eq:divb] are not being solved explicitly but verified via appropriate initial conditions and current deposition procedure. The NSFD differential operators is given by \(\nabla^{*}=D_{x}^{*}\mathbf{\hat{x}}+D_{y}^{*}\mathbf{\hat{y}}+D_{z}^{*}\mathbf{\hat{z}}\) where \(D_{x}^{*}=\left(\alpha+\beta S_{x}^{1}+\xi S_{x}^{2}\right)D_{x}\) with \(S_{x}^{1}G|_{i,j,k}^{n}=G|_{i,j+1,k}^{n}+G|_{i,j-1,k}^{n}+G|_{i,j,k+1}^{n}+G|_{i,j,k-1}^{n}\), \(S_{x}^{2}G|_{i,j,k}^{n}=G|_{i,j+1,k+1}^{n}+G|_{i,j-1,k+1}^{n}+G|_{i,j+1,k-1}^{n}+G|_{i,j-1,k-1}^{n}\). \(G\) is a sample vector component, while \(\alpha\), \(\beta\) and \(\xi\) are constant scalars satisfying \(\alpha+4\beta+4\xi=1\). As with the FDTD algorithm, the quantities with half-integer are located between the nodes (electric field components) or in the center of the cell faces (magnetic field components). The operators along \(y\) and \(z\), i.e. \(D_{y}\), \(D_{z}\), \(D_{y}^{*}\), \(D_{z}^{*}\), \(S_{y}^{1}\), \(S_{z}^{1}\), \(S_{y}^{2}\), and \(S_{z}^{2}\), are obtained by circular permutation of the indices.

Assuming cubic cells (\(\Delta x=\Delta y=\Delta z\)), the coefficients
given in (Karkkainen et al. 2006) (\(\alpha=7/12\), \(\beta=1/12\) and \(\xi=1/48\))
allow for the Courant condition to be at \(\Delta t=\Delta x\), which
equates to having no numerical dispersion along the principal axes.
The algorithm reduces to the FDTD algorithm with \(\alpha=1\) and \(\beta=\xi=0\).
An extension to non-cubic cells is provided by Cowan, *et al.*
in 3-D in (Cowan et al. 2013) and was given by Pukhov in 2-D in
(Pukhov 1999). An alternative NSFDTD implementation that enables superluminous waves is also
given by Lehe et al. in (Lehe et al. 2013).

As mentioned above, a key feature of the algorithms based on NSFDTD is that some implementations (Karkkainen et al. 2006; Cowan et al. 2013) enable the time step \(\Delta t=\Delta x\) along one or more axes and no numerical dispersion along those axes. However, as shown in (Vay et al. 2011), an instability develops at the Nyquist wavelength at (or very near) such a timestep. It is also shown in the same paper that removing the Nyquist component in all the source terms using a bilinear filter (see description of the filter below) suppresses this instability.

### Pseudo Spectral Analytical Time Domain (PSATD)¶

Maxwell’s equations in Fourier space are given by

where \(\tilde{a}\) is the Fourier Transform of the quantity \(a\). As with the real space formulation, provided that the continuity equation \(\partial\tilde{\rho}/\partial t+i\mathbf{k}\cdot\mathbf{\tilde{J}}=0\) is satisfied, then the last two equations will automatically be satisfied at any time if satisfied initially and do not need to be explicitly integrated.

Decomposing the electric field and current between longitudinal and transverse components \(\mathbf{\tilde{E}}=\mathbf{\tilde{E}}_{L}+\mathbf{\tilde{E}}_{T}=\mathbf{\hat{k}}(\mathbf{\hat{k}}\cdot\mathbf{\tilde{E}})-\mathbf{\hat{k}}\times(\mathbf{\hat{k}}\times\mathbf{\tilde{E}})\) and \(\mathbf{\tilde{J}}=\mathbf{\tilde{J}}_{L}+\mathbf{\tilde{J}}_{T}=\mathbf{\hat{k}}(\mathbf{\hat{k}}\cdot\mathbf{\tilde{J}})-\mathbf{\hat{k}}\times(\mathbf{\hat{k}}\times\mathbf{\tilde{J}})\) gives

with \(\mathbf{\hat{k}}=\mathbf{k}/k\).

If the sources are assumed to be constant over a time interval \(\Delta t\), the system of equations is solvable analytically and is given by (see (Haber et al. 1973) for the original formulation and (Jean-Luc Vay, Haber, and Godfrey 2013) for a more detailed derivation):

[Eq:PSATD]

with \(C=\cos\left(k\Delta t\right)\) and \(S=\sin\left(k\Delta t\right)\).

Combining the transverse and longitudinal components, gives

For fields generated by the source terms without the self-consistent dynamics of the charged particles, this algorithm is free of numerical dispersion and is not subject to a Courant condition. Furthermore, this solution is exact for any time step size subject to the assumption that the current source is constant over that time step.

As shown in (Jean-Luc Vay, Haber, and Godfrey 2013), by expanding the coefficients \(S_{h}\) and \(C_{h}\) in Taylor series and keeping the leading terms, the PSATD formulation reduces to the perhaps better known pseudo-spectral time-domain (PSTD) formulation (Dawson 1983; Liu 1997):

The dispersion relation of the PSTD solver is given by \(\sin(\frac{\omega\Delta t}{2})=\frac{k\Delta t}{2}.\) In contrast to the PSATD solver, the PSTD solver is subject to numerical dispersion for a finite time step and to a Courant condition that is given by \(\Delta t\leq \frac{2}{\pi}\left(\frac{1}{\Delta x^{2}}+\frac{1}{\Delta y^{2}}+\frac{1}{\Delta x^{2}}\right)^{-1/2}.\)

The PSATD and PSTD formulations that were just given apply to the field components located at the nodes of the grid. As noted in (Ohmura and Okamura 2010), they can also be easily recast on a staggered Yee grid by multiplication of the field components by the appropriate phase factors to shift them from the collocated to the staggered locations. The choice between a collocated and a staggered formulation is application-dependent.

Spectral solvers used to be very popular in the years 1970s to early 1990s, before being replaced by finite-difference methods with the advent of parallel supercomputers that favored local methods. However, it was shown recently that standard domain decomposition with Fast Fourier Transforms that are local to each subdomain could be used effectively with PIC spectral methods (Jean-Luc Vay, Haber, and Godfrey 2013), at the cost of truncation errors in the guard cells that could be neglected. A detailed analysis of the effectiveness of the method with exact evaluation of the magnitude of the effect of the truncation error is given in (Vincenti and Vay 2016) for stencils of arbitrary order (up-to the infinite “spectral” order).

## Current deposition¶

The current densities are deposited on the computational grid from the particle position and velocities, employing splines of various orders (Abe et al. 1986).

In most applications, it is essential to prevent the accumulation of errors resulting from the violation of the discretized Gauss’ Law. This is accomplished by providing a method for depositing the current from the particles to the grid that preserves the discretized Gauss’ Law, or by providing a mechanism for “divergence cleaning” (Birdsall and Langdon 1991; Langdon 1992; Marder 1987; Vay and Deutsch 1998; Munz et al. 2000). For the former, schemes that allow a deposition of the current that is exact when combined with the Yee solver is given in (Villasenor and Buneman 1992) for linear splines and in (Esirkepov 2001) for splines of arbitrary order.

The NSFDTD formulations given above and in (Pukhov 1999; Vay et al. 2011; Cowan et al. 2013; Lehe et al. 2013) apply to the Maxwell-Faraday equation, while the discretized Maxwell-Ampere equation uses the FDTD formulation. Consequently, the charge conserving algorithms developed for current deposition (Villasenor and Buneman 1992; Esirkepov 2001) apply readily to those NSFDTD-based formulations. More details concerning those implementations, including the expressions for the numerical dispersion and Courant condition are given in (Pukhov 1999; Vay et al. 2011; Cowan et al. 2013; Lehe et al. 2013).

### Current correction¶

In the case of the pseudospectral solvers, the current deposition algorithm generally does not satisfy the discretized continuity equation in Fourier space \(\tilde{\rho}^{n+1}=\tilde{\rho}^{n}-i\Delta t\mathbf{k}\cdot\mathbf{\tilde{J}}^{n+1/2}\). In this case, a Boris correction (Birdsall and Langdon 1991) can be applied in \(k\) space in the form \(\mathbf{\tilde{E}}_{c}^{n+1}=\mathbf{\tilde{E}}^{n+1}-\left(\mathbf{k}\cdot\mathbf{\tilde{E}}^{n+1}+i\tilde{\rho}^{n+1}\right)\mathbf{\hat{k}}/k\), where \(\mathbf{\tilde{E}}_{c}\) is the corrected field. Alternatively, a correction to the current can be applied (with some similarity to the current deposition presented by Morse and Nielson in their potential-based model in (Morse and Nielson 1971)) using \(\mathbf{\tilde{J}}_{c}^{n+1/2}=\mathbf{\tilde{J}}^{n+1/2}-\left[\mathbf{k}\cdot\mathbf{\tilde{J}}^{n+1/2}-i\left(\tilde{\rho}^{n+1}-\tilde{\rho}^{n}\right)/\Delta t\right]\mathbf{\hat{k}}/k\), where \(\mathbf{\tilde{J}}_{c}\) is the corrected current. In this case, the transverse component of the current is left untouched while the longitudinal component is effectively replaced by the one obtained from integration of the continuity equation, ensuring that the corrected current satisfies the continuity equation. The advantage of correcting the current rather than the electric field is that it is more local and thus more compatible with domain decomposition of the fields for parallel computation (Jean Luc Vay, Haber, and Godfrey 2013).

### Vay deposition¶

Alternatively, an exact current deposition can be written for the pseudo-spectral solvers, following the geometrical interpretation of existing methods in real space (Morse and Nielson, 1971; Villasenor and Buneman, 1992; Esirkepov, 2001).

The Vay deposition scheme is the generalization of the Esirkepov deposition scheme for the spectral case with arbitrary-order stencils (Vay et al, 2013). The current density \(\widehat{\boldsymbol{J}}^{\,n+1/2}\) in Fourier space is computed as \(\widehat{\boldsymbol{J}}^{\,n+1/2} = i \, \widehat{\boldsymbol{D}} / \boldsymbol{k}\) when \(\boldsymbol{k} \neq 0\) and set to zero otherwise. The quantity \(\boldsymbol{D}\) is deposited in real space by averaging the currents over all possible grid paths between the initial position \(\boldsymbol{x}^{\,n}\) and the final position \(\boldsymbol{x}^{\,n+1}\) and is defined as

2D Cartesian geometry:

3D Cartesian geometry:

Here, \(w_i\) represents the weight of the \(i\)-th macro-particle and \(\Gamma\) represents its shape factor. Note that in 2D Cartesian geometry, \(D_y\) is effectively \(J_y\) and does not require additional operations in Fourier space.

## Field gather¶

In general, the field is gathered from the mesh onto the macroparticles using splines of the same order as for the current deposition \(\mathbf{S}=\left(S_{x},S_{y},S_{z}\right)\). Three variations are considered:

“momentum conserving”: fields are interpolated from the grid nodes to the macroparticles using \(\mathbf{S}=\left(S_{nx},S_{ny},S_{nz}\right)\) for all field components (if the fields are known at staggered positions, they are first interpolated to the nodes on an auxiliary grid),

“energy conserving (or Galerkin)”: fields are interpolated from the staggered Yee grid to the macroparticles using \(\left(S_{nx-1},S_{ny},S_{nz}\right)\) for \(E_{x}\), \(\left(S_{nx},S_{ny-1},S_{nz}\right)\) for \(E_{y}\), \(\left(S_{nx},S_{ny},S_{nz-1}\right)\) for \(E_{z}\), \(\left(S_{nx},S_{ny-1},S_{nz-1}\right)\) for \(B_{x}\), \(\left(S_{nx-1},S_{ny},S_{nz-1}\right)\) for \(B{}_{y}\) and\(\left(S_{nx-1},S_{ny-1},S_{nz}\right)\) for \(B_{z}\) (if the fields are known at the nodes, they are first interpolated to the staggered positions on an auxiliary grid),

“uniform”: fields are interpolated directly form the Yee grid to the macroparticles using \(\mathbf{S}=\left(S_{nx},S_{ny},S_{nz}\right)\) for all field components (if the fields are known at the nodes, they are first interpolated to the staggered positions on an auxiliary grid).

As shown in :raw-latex:`\cite{BirdsallLangdon,HockneyEastwoodBook,LewisJCP1972}`, the momentum and energy conserving schemes conserve momentum and energy respectively at the limit of infinitesimal time steps and generally offer better conservation of the respective quantities for a finite time step. The uniform scheme does not conserve momentum nor energy in the sense defined for the others but is given for completeness, as it has been shown to offer some interesting properties in the modeling of relativistically drifting plasmas :raw-latex:`\cite{GodfreyJCP2013}`.

# Filtering¶

It is common practice to apply digital filtering to the charge or current density in Particle-In-Cell simulations as a complement or an alternative to using higher order splines (Birdsall and Langdon 1991). A commonly used filter in PIC simulations is the three points filter \(\phi_{j}^{f}=\alpha\phi_{j}+\left(1-\alpha\right)\left(\phi_{j-1}+\phi_{j+1}\right)/2\) where \(\phi^{f}\) is the filtered quantity. This filter is called a bilinear filter when \(\alpha=0.5\). Assuming \(\phi=e^{jkx}\) and \(\phi^{f}=g\left(\alpha,k\right)e^{jkx}\), the filter gain \(g\) is given as a function of the filtering coefficient \(\alpha\) and the wavenumber \(k\) by \(g\left(\alpha,k\right)=\alpha+\left(1-\alpha\right)\cos\left(k\Delta x\right)\approx1-\left(1-\alpha\right)\frac{\left(k\Delta x\right)^{2}}{2}+O\left(k^{4}\right)\). The total attenuation \(G\) for \(n\) successive applications of filters of coefficients \(\alpha_{1}\)…\(\alpha_{n}\) is given by \(G=\prod_{i=1}^{n}g\left(\alpha_{i},k\right)\approx1-\left(n-\sum_{i=1}^{n}\alpha_{i}\right)\frac{\left(k\Delta x\right)^{2}}{2}+O\left(k^{4}\right)\). A sharper cutoff in \(k\) space is provided by using \(\alpha_{n}=n-\sum_{i=1}^{n-1}\alpha_{i}\), so that \(G\approx1+O\left(k^{4}\right)\). Such step is called a “compensation” step (Birdsall and Langdon 1991). For the bilinear filter (\(\alpha=1/2\)), the compensation factor is \(\alpha_{c}=2-1/2=3/2\). For a succession of \(n\) applications of the bilinear factor, it is \(\alpha_{c}=n/2+1\).

It is sometimes necessary to filter on a relatively wide band of wavelength, necessitating the application of a large number of passes of the bilinear filter or on the use of filters acting on many points. The former can become very intensive computationally while the latter is problematic for parallel computations using domain decomposition, as the footprint of the filter may eventually surpass the size of subdomains. A workaround is to use a combination of filters of limited footprint. A solution based on the combination of three point filters with various strides was proposed in (Vay et al. 2011) and operates as follows.

The bilinear filter provides complete suppression of the signal at the grid Nyquist wavelength (twice the grid cell size). Suppression of the signal at integer multiples of the Nyquist wavelength can be obtained by using a stride \(s\) in the filter \(\phi_{j}^{f}=\alpha\phi_{j}+\left(1-\alpha\right)\left(\phi_{j-s}+\phi_{j+s}\right)/2\) for which the gain is given by \(g\left(\alpha,k\right)=\alpha+\left(1-\alpha\right)\cos\left(sk\Delta x\right)\approx1-\left(1-\alpha\right)\frac{\left(sk\Delta x\right)^{2}}{2}+O\left(k^{4}\right)\). For a given stride, the gain is given by the gain of the bilinear filter shifted in k space, with the pole \(g=0\) shifted from the wavelength \(\lambda=2/\Delta x\) to \(\lambda=2s/\Delta x\), with additional poles, as given by \(sk\Delta x=\arccos\left(\frac{\alpha}{\alpha-1}\right)\pmod{2\pi}\). The resulting filter is pass band between the poles, but since the poles are spread at different integer values in k space, a wide band low pass filter can be constructed by combining filters using different strides. As shown in (Vay et al. 2011), the successive application of 4-passes + compensation of filters with strides 1, 2 and 4 has a nearly equivalent fall-off in gain as 80 passes + compensation of a bilinear filter. Yet, the strided filter solution needs only 15 passes of a three-point filter, compared to 81 passes for an equivalent n-pass bilinear filter, yielding a gain of 5.4 in number of operations in favor of the combination of filters with stride. The width of the filter with stride 4 extends only on 9 points, compared to 81 points for a single pass equivalent filter, hence giving a gain of 9 in compactness for the stride filters combination in comparison to the single-pass filter with large stencil, resulting in more favorable scaling with the number of computational cores for parallel calculations.

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