Convergence of a semi-Lagrangian scheme for the ellipsoidal BGK model of the Boltzmann equation

TL;DR

This paper introduces a new implicit semi-Lagrangian scheme for the ellipsoidal BGK model of the Boltzmann equation, providing stability, efficiency, and uniform convergence estimates across all relaxation parameters.

Contribution

The paper develops a novel semi-Lagrangian scheme for the ellipsoidal BGK model with proven uniform convergence, including the original BGK case.

Findings

Scheme guarantees stability and efficiency.

Error estimate derived in weighted $L^{inity}$ norm.

Convergence holds uniformly for all relaxation parameters.

Abstract

The ellipsoidal BGK model is a generalized version of the original BGK model designed to reproduce the physical Prandtl number in the Navier-Stokes limit. In this paper, we propose a new implicit semi-Lagrangian scheme for the ellipsoidal BGK model, which, by exploiting special structures of the ellipsoidal Gaussian, can be transformed into a semi-explicit form, guaranteeing the stability of the implicit methods and the efficiency of the explicit methods at the same time. We then derive an error estimate of this scheme in a weighted $L^{\infty}$ norm. Our convergence estimate holds uniformly in the whole range of relaxation parameter $\nu$ including $\nu=0$, which corresponds to the original BGK model.