Conservative semi-Lagrangian schemes for kinetic equations Part II: Applications

TL;DR

This paper introduces high-order conservative semi-Lagrangian schemes for kinetic equations, demonstrating their accuracy, stability, and asymptotic preserving properties through applications to various models like Vlasov-Poisson and BGK.

Contribution

It extends previous work by developing high-order, conservative semi-Lagrangian methods applicable to multiple kinetic models with proven accuracy and asymptotic preservation.

Findings

Schemes are high order accurate in space and time.

Methods preserve positivity and the L1-norm for initial positive solutions.

Schemes are asymptotic preserving and robust in test cases.

Abstract

In this paper, we present a new class of conservative semi-Lagrangian schemes for kinetic equations. They are based on the conservative reconstruction technique introduced in [S. Y. Cho, et al., Conservative semi-Lagrangian schemes for kinetic equations Part I: Reconstruction, 2020]. The methods are high order accurate both in space and time. Because of the semi-Lagrangian nature, the time step is not restricted by a CFL-type condition. Applications are presented to rigid body rotation, Vlasov Poisson system and the BGK model of rarefied gas dynamics. In the first two cases operator splitting is adopted, to obtain high order accuracy in time, and a conservative reconstruction that preserves the maximum and minimum of the function is used. For initially positive solutions, in particular, this guarantees exact preservation of the L1-norm. Conservative schemes for the BGK model are constructed by coupling the conservative reconstruction with a conservative treatment of the collision term. High order in time is obtained by either Runge-Kutta or BDF time discretization of the equation along characteristics. Because of L-stability and exact conservation, the resulting scheme for the BGK model is asymptotic preserving for the underlying fluid dynamic limit. Several test cases in one and two space dimensions confirm the accuracy and robustness of the methods, and the AP property of the schemes when applied to the BGK model.