An Energy-Conserving Fourier Particle-in-Cell Method with Asymptotic-Preserving Preconditioner for Vlasov-Amp\`ere System with Exact Curl-Free Constraint

TL;DR

This paper introduces an energy-conserving implicit particle-in-cell algorithm for the Vlasov-Ampère system, utilizing a Fourier discretization and an asymptotic-preserving preconditioner to accurately simulate complex plasma systems across multiple scales.

Contribution

The paper develops a novel structure-preserving Fourier PIC method with an asymptotic-preserving preconditioner, ensuring energy conservation and robustness for multiscale plasma simulations.

Findings

The proposed method accurately conserves energy in plasma simulations.

It outperforms classical schemes in multiscale plasma systems.

Numerical experiments validate the method's robustness and efficiency.

Abstract

We present an efficient and accurate energy-conserving implicit particle-in-cell~(PIC) algorithm for the electrostatic Vlasov system, with particular emphasis on its high robustness for simulating complex plasma systems with multiple physical scales. This method consists of several indispensable elements: (\romannumeral1) the reformulation of the original Vlasov-Poisson system into an equivalent Vlasov-Amp\`ere system with divergence-/curl-free constraints; (\romannumeral2) a novel structure-preserving Fourier spatial discretization, which exactly preserves these constraints at the discrete level; (\romannumeral3) a preconditioned Anderson-acceleration algorithm for the solution of the highly nonlinear system; and (\romannumeral4) a linearized and uniform approximation of the implicit Crank-Nicolson scheme for various Debye lengths, based on the generalized Ohm's law, which serves as an asymptotic-preserving preconditioner for the proposed method. Numerical experiments are conducted, and comparisons are made among the proposed energy-conserving scheme, the classical leapfrog scheme, and a Strang operator-splitting scheme to demonstrate the superiority of the proposed method, especially for plasma systems crossing physical scales.