Conservative Multi-Dimensional Semi-Lagrangian Finite Difference Scheme: Stability and Applications to the Kinetic and Fluid Simulations

TL;DR

This paper introduces a mass conservative semi-Lagrangian finite difference scheme for multi-dimensional problems, ensuring local mass conservation and stability, and demonstrates its effectiveness on kinetic and fluid simulations.

Contribution

It presents a novel conservative correction procedure for semi-Lagrangian schemes that guarantees mass conservation without dimensional splitting.

Findings

The scheme is stable under certain time step constraints.

Numerical tests confirm the scheme's effectiveness on linear and nonlinear problems.

The method successfully applies to Vlasov-Poisson and Euler systems.

Abstract

In this paper, we propose a mass conservative semi-Lagrangian finite difference scheme for multi-dimensional problems without dimensional splitting. The semi-Lagrangian scheme, based on tracing characteristics backward in time from grid points, does not necessarily conserve the total mass. To ensure mass conservation, we propose a conservative correction procedure based on a flux difference form. Such procedure guarantees local mass conservation, while introducing time step constraints for stability. We theoretically investigate such stability constraints from an ODE point of view by assuming exact evaluation of spatial differential operators and from the Fourier analysis for linear PDEs. The scheme is tested by classical two dimensional linear passive-transport problems, such as linear advection, rotation and swirling deformation. The scheme is applied to solve the nonlinear Vlasov-Poisson system using a a high order tracing mechanism proposed in [Qiu and Russo, 2016]. Such high order characteristics tracing scheme is generalized to the nonlinear guiding center Vlasov model and incompressible Euler system. The effectiveness of the proposed conservative semi-Lagrangian scheme is demonstrated numerically by our extensive numerical tests.