Explicit relaxation Particle-in-Cell methods for Vlasov-Poisson equations with a strong magnetic field

TL;DR

This paper introduces explicit relaxation Particle-in-Cell methods for Vlasov-Poisson equations with strong magnetic fields, achieving energy conservation and rigorous error bounds.

Contribution

The work develops a new family of ER-PIC schemes that combine splitting and relaxation parameter updates, ensuring energy conservation and proven accuracy.

Findings

Exact energy conservation achieved by the schemes.

Second-order error bounds for the Strang-type method.

First-order accuracy in position for the Lie-Trotter scheme.

Abstract

In this work, we present a novel family of explicit relaxation Particle-in-Cell (ER-PIC) methods for the Vlasov-Poisson equation with a strong magnetic field. These schemes achieve exact energy conservation by combining a splitting framework with the dynamic updating of a relaxation parameter at each time step. Using an averaging technique, we rigorously establish second-order error bounds for the Strang-type ER-PIC method and uniform first order accuracy in position for the Lie-Trotter ER-PIC scheme. Numerical experiments across the fluid, finite Larmor radius, and diffusion regimes confirm the accuracy and energy conservation of our methods.