Jin-Xin relaxation as a shock-capturing method for high-order DG/FR schemes
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arXiv
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| Format: | Preprint |
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2026
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| author | Artiano, Marco Babbar, Arpit Schlottke-Lakemper, Michael Gassner, Gregor Ranocha, Hendrik |
| author_facet | Artiano, Marco Babbar, Arpit Schlottke-Lakemper, Michael Gassner, Gregor Ranocha, Hendrik |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_16290 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Jin-Xin relaxation as a shock-capturing method for high-order DG/FR schemes Artiano, Marco Babbar, Arpit Schlottke-Lakemper, Michael Gassner, Gregor Ranocha, Hendrik Numerical Analysis 65M60 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. |
| title | Jin-Xin relaxation as a shock-capturing method for high-order DG/FR schemes |
| topic | Numerical Analysis 65M60 |
| url | https://arxiv.org/abs/2603.16290 |