Jin-Xin relaxation as a shock-capturing method for high-order DG/FR schemes

TL;DR

This paper introduces a shock-capturing approach for high-order DG/FR schemes using Jin-Xin relaxation, which adapts dissipation based on local smoothness to improve numerical stability and accuracy.

Contribution

It proposes a novel method that employs Jin-Xin relaxation with adaptive b5 to enhance shock capturing in high-order DG/FR schemes using IMEX-RK methods.

Findings

Effective shock capturing demonstrated on Burgers' and Euler equations.

Adaptive b5 improves stability in non-smooth regions.

Method integrates smoothly with high-order schemes.

Abstract

Jin-Xin relaxation is a method for approximating non-linear hyperbolic conservation laws by a linear system of hyperbolic equations with an $\varepsilon$ dependent stiff source term. The system formally relaxes to the original conservation law as $\varepsilon \to 0$. An asymptotic analysis of the Jin-Xin relaxation system shows that it can be seen as a convection-diffusion equation with a diffusion coefficient that depends on the relaxation parameter $\varepsilon$. This work makes use of this property to use the Jin-Xin relaxation system as a shock-capturing method for high-order discontinuous Galerkin (DG) or flux reconstruction (FR) schemes. The idea is to use a smoothness indicator to choose the $\varepsilon$ value in each cell, so that we can use larger $\varepsilon$ values in non-smooth regions to add extra numerical dissipation. We show how this can be done by using a single stage method by using the compact Runge-Kutta FR method that handles the stiff source term by using IMplicit-EXplicit Runge-Kutta (IMEX-RK) schemes. Numerical results involving Burgers' equation and the compressible Euler equations are shown to demonstrate the effectiveness of the proposed method.