Uniform accuracy of implicit-explicit backward differentiation formulas (IMEX-BDF) for linear hyperbolic relaxation systems

TL;DR

This paper proves that certain IMEX-BDF schemes maintain uniform stability and accuracy across all regimes for linear hyperbolic relaxation systems, regardless of the relaxation time, verified through numerical experiments.

Contribution

It establishes the uniform stability and accuracy of IMEX-BDF schemes for hyperbolic relaxation systems, independent of relaxation time, with spectral spatial discretization.

Findings

Uniform stability and accuracy proven for IMEX-BDF schemes

Accuracy independent of relaxation time in all regimes

Numerical validation on traffic flows, gas dynamics, and kinetic theory

Abstract

This work is concerned with the uniform accuracy of implicit-explicit backward differentiation formulas for general linear hyperbolic relaxation systems satisfying the structural stability condition proposed previously by the third author. We prove the uniform stability and accuracy of a class of IMEX-BDF schemes discretized spatially by a Fourier spectral method. The result reveals that the accuracy of the fully discretized schemes is independent of the relaxation time in all regimes. It is verified by numerical experiments on several applications to traffic flows, rarefied gas dynamics and kinetic theory.