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

Marco Artiano, Arpit Babbar, Michael Schlottke-Lakemper, Gregor Gassner, Hendrik Ranocha

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.

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

Abstract

Jin-Xin relaxation is a method for approximating non-linear hyperbolic conservation laws by a linear system of hyperbolic equations with an dependent stiff source term. The system formally relaxes to the original conservation law as . 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 . 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 value in each cell, so that we can use larger 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.
Paper Structure (13 sections, 34 equations, 5 figures)

This paper contains 13 sections, 34 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Compact Runge-Kutta FR method for stiff source terms
  3. Review of the Jin-Xin relaxation system
  4. Shock capturing with Jin-Xin relaxation
  5. Numerical results
  6. Scalar equations
  7. Burgers' equation: convex flux
  8. Buckley-Leverett equation: non-convex flux
  9. Euler's equations
  10. Woodward-Colella blast wave problem
  11. Sedov's blast problem
  12. Kelvin-Helmholtz instability
  13. Conclusions and future work

Figures (5)

  • Figure 1: (a) Piece-wise polynomial solution at time $t_n$, and (b) discontinuous and continuous flux. The figure is inspired from babbar2022lax.
  • Figure 2: Comparison of subcell-based blending limiter and the Jin-Xin relaxation based limiter for (a) Burgers' equation and (b) Buckley-Leverett equation. The SSP3-IMEX(4,3,3) scheme of pareschi2005 is used for the Jin-Xin relaxation method.
  • Figure 3: Comparison of subcell-based blending limiter and the Jin-Xin relaxation based limiter for the blast wave problem showing (a) Density and (b) Pressure. The SSP3-IMEX(4,3,3) scheme of pareschi2005 is used for the Jin-Xin relaxation method.
  • Figure 4: Comparison of subcell-based blending limiter and the Jin-Xin relaxation based shock capturing scheme for the Sedov's blast wave problem. The BPR(3, 4, 3) scheme of boscarino2013 is used for the Jin-Xin relaxation method.
  • Figure 5: Comparison of subcell-based blending limiter and the Jin-Xin relaxation based shock capturing scheme for the Kelvin-Helmholtz instability problem. The BPR(3, 4, 3) scheme of boscarino2013 is used for the Jin-Xin relaxation method.