A conservative semi-Lagrangian scheme for the ellipsoidal BGK model of the Boltzmann equation

TL;DR

This paper introduces a high-order conservative semi-Lagrangian scheme for the ellipsoidal BGK model, effectively handling stiff relaxation and convection without Newton solvers, and demonstrating accuracy for Navier-Stokes and Boltzmann equations.

Contribution

It develops a novel semi-Lagrangian method with implicit time discretization that avoids Newton solvers, improving efficiency and accuracy for kinetic equations.

Findings

Accurate solutions for Navier-Stokes and Boltzmann equations at moderate Knudsen numbers.

The scheme effectively handles stiff relaxation operators with high order stability.

Numerical tests confirm the method's efficiency and precision.

Abstract

In this paper, we propose a high order conservative semi-Lagrangian scheme (SL) for the ellipsoidal BGK model of the Boltzmann transport equation. To avoid the time step restriction induced by the convection term, we adopt the semi-Lagrangian approach. For treating the nonlinear stiff relaxation operator with small Knudsen number, we employ high order $L$-stable diagonally implicit Runge-Kutta time discretization or backward difference formula. The proposed implicit schemes are designed to update solutions explicitly without resorting to any Newton solver. We present several numerical tests to demonstrate the accuracy and efficiency of the proposed methods. These methods allow us to obtain accurate approximations of the solutions to the Navier-Stokes equations or the Boltzmann equation for moderate or relatively large Knudsen numbers, respectively.