Asymptotic Analysis of IMEX-RK Methods for ES-BGK Model at Navier-Stokes level

TL;DR

This paper extends the asymptotic analysis of IMEX-RK methods for the ES-BGK model at the Navier-Stokes level, demonstrating their ability to accurately capture fluid dynamics across different regimes.

Contribution

It generalizes the asymptotic analysis to Type I and Type II IMEX-RK schemes and proposes methods for uniform accuracy across Knudsen numbers.

Findings

IMEX-RK schemes can accurately capture Navier-Stokes limit

Proposed methods achieve uniform accuracy across regimes

Numerical examples verify theoretical results

Abstract

Implicit-explicit Runge-Kutta (IMEX-RK) time discretization methods are very popular when solving stiff kinetic equations. In [21], an asymptotic analysis shows that a specific class of high-order IMEX-RK schemes can accurately capture the Navier-Stokes limit without needing to resolve the small scales dictated by the Knudsen number. In this work, we extend the asymptotic analysis to general IMEX-RK schemes, known in literature as Type I and Type II. We further suggest some IMEX-RK methods developed in the literature to attain uniform accuracy in the wide range of Knudsen numbers. Several numerical examples are presented to verify the validity of the obtained theoretical results and the effectiveness of the methods.