Formulation of the relativistic moment implicit particle-in-cell method

Abstract

A new formulation is presented for the implicit moment method applied to the time-dependent relativistic Vlasov-Maxwell system. The new approach is based on a specific formulation of the implicit moment method that allows us to retain the same formalism that is valid in the classical case despite the formidable complication introduced by the nonlinear nature of the relativistic equations of motion. To demonstrate the validity of the new formulation, an implicit finite difference algorithm is developed to solve the Maxwell’s equations and equations of motion. A number of benchmark problems are run: two stream instability, ion acoustic wave damping, Weibel instability, and Poynting flux acceleration. The numerical results are all in agreement with analytical solutions.

Particle-in-cell modeling of relativistic laser–plasma interaction with the adjustable-damping, direct implicit method

“…[19] in the electrostatic case, it then appears that the direct implicit method can be derived as a one-iteration Newton method with the starting values X ð0Þ ¼ e X nþ1 ; U ð0Þ ¼ e U nþ1=2 and E ð0Þ ¼ 0. An alternative wave equation has been recently derived in the relativistic regime in the framework of the moment implicit method [17]. As shown in Appendix A, the major difference between the direct and moment implicit methods stems from the linearization of the current density.…”

“…To this goal, we shall adopt the notations recently used by Noguchi et al [17] to derive the relativistic formulation of the moment implicit method. The major difference between the direct and moment methods stems from the linearization of the current density.…”

“…The first one to be published, the so-called moment method, makes use of the fluid equations to predict future source terms [11][12][13][14][15][16]. and has been recently extended to the relativistic regime [17]. The present article will focus on the alternate approach, referred to as the direct implicit method, which is based on a direct linearization of the Lorentz equations [18][19][20][21].…”

A Comprehensive Comparison of Relativistic Particle Integrators

“…Pétri (2017) presents a fully implicit scheme similar to the one above, but with a different choice of average velocity (specifically, the one used in Vay 2008). The solution of system (24) can be carried out using several approaches (see, e.g., Noguchi et al 2007). In this work, we choose to adopt a Newton algorithm similar to the one presented in Siddi et al (2017) but extended here to the relativistic case.…”

“…To formally prove energy conservation for our implicit particle mover, we repeat the argument of Noguchi et al (2007) for the relativistic equation of motion. Starting from the discretized equation of motion, where n and n 1 + indicate consecutive time levels,…”