A domain decomposition method for pseudo-spectral electromagnetic simulations of plasmas

doi:10.1016/j.jcp.2013.03.010

Journal of Computational Physics

Volume 243, 15 June 2013, Pages 260-268

https://doi.org/10.1016/j.jcp.2013.03.010 Get rights and content

Abstract

Pseudo-spectral electromagnetic solvers (i.e. representing the fields in Fourier space) have extraordinary precision. In particular, Haber et al. presented in 1973 a pseudo-spectral solver that integrates analytically the solution over a finite time step, under the usual assumption that the source is constant over that time step. Yet, pseudo-spectral solvers have not been widely used, due in part to the difficulty for efficient parallelization owing to global communications associated with global FFTs on the entire computational domains.

A method for the parallelization of electromagnetic pseudo-spectral solvers is proposed and tested on single electromagnetic pulses, and on Particle-In-Cell simulations of the wakefield formation in a laser plasma accelerator.

The method takes advantage of the properties of the Discrete Fourier Transform, the linearity of Maxwell’s equations and the finite speed of light for limiting the communications of data within guard regions between neighboring computational domains.

Although this requires a small approximation, test results show that no significant error is made on the test cases that have been presented.

The proposed method opens the way to solvers combining the favorable parallel scaling of standard finite-difference methods with the accuracy advantages of pseudo-spectral methods.

Introduction

Particle-In-Cell (PIC) has been the method of choice for the last fifty years for modeling plasmas that include kinetic effects. The most popular electromagnetic formulation uses finite difference discretization of Maxwell’s equations in both space and time (FDTD), which produces fast solvers that scale well in parallel, but suffers from various anomalous numerical effects resulting from discretization, field staggering, and numerical dispersion. Aliasing and numerical dispersion lead to an unphysical numerical Cherenkov instability when relativistic particles interact with their gridded self-field [1]. An example is a strong instability that appears in simulations of Laser-Plasma Acceleration (LPA) in Lorentz boosted frames [2], [3] or astrophysical shocks [4]. In addition, the staggering of electric and magnetic fields leads to inexact cancelation of relativistic beams’ self electric and magnetic components in the calculation of the Lorentz force $E + v \times B$ [5].

Pseudo-spectral methods, which advance the fields in Fourier space [6], offer a number of advantages over standard FDTD solvers. An analytical solution for Maxwell’s equations on a grid was given by Haber et al. [7] for a periodic system, leading to a solver that is accurate to machine precision for the modes resolved by the calculation grid, that has no Courant time step limit in vacuum and that has no numerical dispersion. Furthermore, pseudo-spectral methods naturally represent all field values at the nodes of a grid, eliminating staggering errors.

Due to global communications associated with the use of Fast Fourier Transforms (FFTs) that span the entire domain, pseudo-spectral electromagnetic solvers have not scaled beyond a few thousands of cores. In contrast, the FDTD solvers allow parallelization requiring only local communications between neighboring subdomains, and thus excellent scaling to hundreds of thousand of CPU cores.

In this communication, we report on a domain decomposition method that enables parallelization of pseudo-spectral solvers and requires only local FFTs and communications between neighboring subdomains. This is similar to FDTD decomposition, potentially allowing more accurate pseudo-spectral methods to be scaled to the same level as FDTD. Although this decomposition requires a small approximation, as discussed below, test results show that the error introduced is sufficiently small in practice for the cases that have been tested.

We first present several variations of pseudo-spectral solvers, followed by the new method for domain decomposition, and its application to pseudo-spectral PIC simulations. An example of the application of the method is given on the modeling of the wakefield formation in a laser plasma accelerator [8].

Section snippets

Pseudo-spectral electromagnetic solvers

Maxwell’s equations in Fourier space are given by $\frac{\partial \tilde{E}}{\partial t} = ic k \times \tilde{B} - \tilde{J},$ $\frac{\partial \tilde{B}}{\partial t} = - ic k \times \tilde{E},$ $i k \cdot \tilde{E} = \tilde{ρ},$ $i k \cdot \tilde{B} = 0,$ where $\tilde{a}$ is the Fourier Transform of the quantity a. Provided that the continuity equation $\partial \tilde{ρ} / \partial t + i k \cdot \tilde{J} = 0$ 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 $\tilde{E} = {\tilde{E}}_{L} + {\tilde{E}}_{T} = \hat{k} (\hat{k} \cdot \tilde{E}) - \hat{k} \times \hat{k} \times \tilde{E}$ and $\tilde{J} = {\tilde{J}}_{L} + {\tilde{J}}_{T} = \hat{k} (\hat{k} \cdot \tilde{J}) -$

Parallelization using domain decomposition

In this section, the properties of the Discrete Fourier Transform (DFT), as well as the linearity of Maxwell’s equations and the finite speed of light are exploited to enable parallelization of electromagnetic pseudo-spectral solvers using local FFTs, with domain decomposition and communications via guard cells.

The method is based on the three following premises:

1.
the properties of DFTs ensure that at any given time step, each electromagnetic field component discretized on a regular orthogonal

Conclusion

Pseudo-spectral electromagnetic solvers (i.e. representing the fields in Fourier space) have a number of advantages over standard FDTD solvers. Yet, they have not been widely used, due in part to the difficulty for efficient parallelization owing to global communications associated with global FFTs.

A method for the parallelization of electromagnetic pseudo-spectral solvers has been presented that requires only local FFTs and exchange of local guard cell data between neighboring regions. It has

Acknowledgments

We are thankful to C.G.R. Geddes, D.P. Grote and A. Friedman for valuable comments. Work was supported in part by US-DOE Contracts DE-AC02-05CH11231 and US-DOE SciDAC program ComPASS.

This document was prepared as an account of work sponsored in part by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor The Regents of the University of California, nor any of their employees, makes any

References (18)

B. Godfrey
Numerical Cherenkov instabilities in electromagnetic particle codes
Journal of Computational Physics
(1974)
J. Villasenor et al.
Rigorous charge conservation for local electromagnetic-field solvers
Computer Physics Communications
(1992)
T. Esirkepov
Exact charge conservation scheme for particle-in-cell simulation with an arbitrary form-factor
Computer Physics Communications
(2001)
C. Munz et al.
Divergence correction techniques for Maxwell solvers based on a hyperbolic model
Journal of Computational Physics
(2000)
J.L. Vay et al.
Numerical methods for instability mitigation in the modeling of laser wakefield accelerators in a Lorentz-boosted frame
Journal Of Computational Physics
(2011)
J.L. Vay
Noninvariance of space- and time-scale ranges under a Lorentz transformation and the implications for the study of relativistic interactions
Physical Review Letters
(2007)
J.L. Vay, C.G.R. Geddes, E. Esarey, C.B. Schroeder, W.P. Leemans, E. Cormier-Michel, D.P. Grote, Modeling of 10Gev–1Tev...
L. Sironi, A. Spitkovsky, Private Communication,...
J.L. Vay
Simulation of beams or plasmas crossing at relativistic velocity
Physics of Plasmas
(2008)

There are more references available in the full text version of this article.

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Short NoteA domain decomposition method for pseudo-spectral electromagnetic simulations of plasmas

Abstract

Introduction

Section snippets

Pseudo-spectral electromagnetic solvers

Parallelization using domain decomposition

Conclusion

Acknowledgments

Journal of Computational Physics

Computer Physics Communications

Computer Physics Communications

Journal of Computational Physics

Journal Of Computational Physics

Noninvariance of space- and time-scale ranges under a Lorentz transformation and the implications for the study of relativistic interactions

Physical Review Letters

Simulation of beams or plasmas crossing at relativistic velocity

Physics of Plasmas

Short Note
A domain decomposition method for pseudo-spectral electromagnetic simulations of plasmas