A WENO-based Method of Line Transpose Approach for Vlasov Simulations

TL;DR

This paper introduces a high-order implicit WENO-based Method of Line Transpose approach for Vlasov simulations, achieving accurate, stable, and larger time steps while preserving positivity and handling discontinuities effectively.

Contribution

It develops a novel high-order implicit MOL$^T$ scheme with WENO and positivity-preserving limiter for Vlasov simulations, improving stability and accuracy over explicit methods.

Findings

High order accuracy in space and time.

Ability to take larger time steps than explicit schemes.

Effective handling of discontinuities and sharp gradients.

Abstract

In this paper, a high order implicit Method of Line Transpose (MOL$^T$ ) method based on a weighted essentially non-oscillatory (WENO) methodology is developed for one-dimensional linear transport equations and further applied to the Vlasov-Poisson (VP) simulations via dimensional splitting. In the MOL$^T$ framework, the time variable is first discretized by a diagonally implicit strong-stability-preserving Runge-Kutta method, resulting in a boundary value problem (BVP) at the discrete time levels. Then an integral formulation coupled with a high order WENO methodology is employed to solve the BVP. As a result, the proposed scheme is high order accurate in both space and time and free of oscillations even though the solution is discontinuous or has sharp gradients. Moreover, the scheme is able to take larger time step evolution compared with an explicit MOL WENO scheme with the same order of accuracy. The desired positivity-preserving (PP) property of the scheme is further attained by incorporating a newly proposed high order PP limiter. We perform numerical experiments on several benchmarks including linear advection, solid body rotation problem; and on the Landau damping, two-stream instabilities, bump-on-tail, and plasma sheath by solving the VP system. The efficacy and efficiency of the proposed scheme is numerically verified.