Jin-Xin relaxation as a shock-capturing method for high-order DG/FR schemes
This addresses shock-capturing challenges in computational fluid dynamics for researchers, but it is incremental as it adapts an existing relaxation method to high-order schemes.
The paper tackled the problem of shock-capturing in high-order discontinuous Galerkin or flux reconstruction schemes for hyperbolic conservation laws by using the Jin-Xin relaxation system with adaptive ε values, resulting in effective numerical dissipation in non-smooth regions as demonstrated with Burgers' equation and compressible Euler equations.
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.