A high-order weighted positive and flux conservative method for the Vlasov equation

TL;DR

This paper introduces a high-order, conservative, positivity-preserving numerical scheme for the Vlasov equation that improves accuracy and spectral properties, enabling high-resolution plasma simulations.

Contribution

A novel fifth-order scheme combining positivity, non-oscillatory features, and enhanced spectral performance for solving Vlasov equations.

Findings

Achieves formal fifth-order accuracy.

Outperforms lower and higher order schemes in spectral properties.

Demonstrates high-resolution simulations with better entropy conservation.

Abstract

We present a high-order conservative, positivity-preserving, and non-oscillatory scheme for solving the Vlasov equation. The scheme attains formal fifth-order accuracy through a convex combination of positive and non-oscillatory polynomials in substencils. Nonlinear weights for these polynomials are formulated that assign higher priority to substencils with larger L2 norm to enhance resolution while maintaining positivity and non-oscillatory properties. An approximate dispersion relation indicates that the spectral properties of the present scheme outperform those of an underlying fifth-order scheme and even surpass those of a seventh-order scheme in certain wavenumber ranges. We apply this scheme to the one-dimensional Vlasov-Ampere equations and the two-dimensional Vlasov-Maxwell equations, and demonstrate high-resolution simulations with improved conservation of entropy.