Convergence of a Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation

TL;DR

This paper proves the convergence of a semi-Lagrangian numerical scheme for the BGK model of the Boltzmann equation, demonstrating that the discrete solution approaches the smooth solution with an explicit error bound.

Contribution

It provides the first rigorous convergence proof and explicit error estimate for a semi-Lagrangian scheme applied to the BGK model of the Boltzmann equation.

Findings

Discrete solution converges in weighted L1 norm

Explicit error estimate derived

Scheme is stable without CFL restriction

Abstract

Recently, a new class of semi-Lagrangian methods for the BGK model of the Boltzmann equation has been introduced [8, 17, 18]. These methods work in a satisfactory way either in rarefied or fluid regime. Moreover, because of the semi-Lagrangian feature, the stability property is not restricted by the CFL condition. These aspects make them very attractive for practical applications. In this paper, we investigate the convergence properties of the method and prove that the discrete solution of the scheme converges in a weighted L1 norm to the unique smooth solution by deriving an explicit error estimate.