A split-step Active Flux method for the Vlasov-Poisson system

TL;DR

This paper introduces a split-step Active Flux method tailored for the Vlasov-Poisson system, enabling high-order, conservative numerical solutions for plasma evolution in higher dimensions.

Contribution

It extends Active Flux to the Vlasov-Poisson system with dimensional splitting and three flux integral formulations, enhancing high-order accuracy in plasma simulations.

Findings

Numerical results demonstrate the effectiveness of the proposed methods.

Third-order reconstructions outperform second-order in accuracy.

The discrepancy formulation provides a robust high-order approximation.

Abstract

Active Flux is a modified Finite Volume method that evolves additional Degrees of Freedom for each cell that are located on the interface by a non-conservative method to compute high-order approximations to the numerical fluxes through the respective interface to evolve the cell-average in a conservative way. In this paper, we apply the method to the Vlasov-Poisson system describing the time evolution of the time-dependent distribution function of a collisionless plasma. In particular, we consider the evaluation of the flux integrals in higher dimensions. We propose a dimensional splitting and three types of formulations of the flux integral: a one-dimensional reconstruction of second order, a third-order reconstruction based on information along each dimension, and a third-order reconstruction based on a discrepancy formulation of the Active Flux method. Numerical results in 1D1V phase-space compare the properties of the various methods.