New Preprint ‘Jin-Xin relaxation as a shock-capturing method for high-order DG/FR schemes’ on arXiv

Marco Artiano, Arpit Babbar, Michael Schlottke-Lakemper, Gregor Gassner, and I have published our new preprint Jin-Xin relaxation as a shock-capturing method for high-order DG/FR schemes on arXiv.

Jin-Xin relaxation is a method for approximating non-linear hyperbolic conservation laws by a linear system of hyperbolic equations with an ε dependent stiff source term. The system formally relaxes to the original conservation law as ε→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 ε. 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 ε value in each cell, so that we can use larger ε 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.

The reproducibility repository is available on GitHub and Zenodo.