Uniform accuracy of implicit-explicit Runge-Kutta methods for linear hyperbolic relaxation systems

TL;DR

This paper proves that certain IMEX Runge-Kutta methods maintain consistent accuracy across different regimes for linear hyperbolic relaxation systems, with validation through numerical experiments.

Contribution

It establishes the uniform stability and accuracy of a class of IMEX-RK schemes for hyperbolic relaxation systems, independent of relaxation time.

Findings

Accuracy is independent of relaxation time across regimes

Numerical experiments confirm theoretical stability and accuracy

Applicable to traffic flow and kinetic theory models

Abstract

In this paper, we study the uniform accuracy of implicit-explicit (IMEX) Runge-Kutta (RK) schemes for general linear hyperbolic relaxation systems satisfying the structural stability condition proposed in \cite{yong_singular_1999}. We establish the uniform stability and accuracy of a class of IMEX-RK schemes with spatial discretization using a Fourier spectral method. Our results demonstrate that the accuracy of the fully discretized schemes is independent of the relaxation time across all regimes. Numerical experiments on applications in traffic flows and kinetic theory verify our theoretical analysis.