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Asymptotic preserving schemes for hyperbolic systems with relaxation

C Mahmoud, H Mathis

TL;DR

The paper addresses numerical approximation of hyperbolic systems with relaxation by introducing two asymptotic-preserving schemes that solve the full relaxation system without splitting the convective and source terms. The first scheme embeds the centered FORCE flux within an unsplit staggered framework, while the second uses an approximate Riemann solver that accounts for the source term, both designed to converge to the equilibrium model as $\varepsilon\to0$ and to the fully non-relaxed hyperbolic system as $\varepsilon\to\infty$. For specific models, invariant-domain preservation and a discrete entropy inequality are established (notably for Jin–Xin), and extensive numerical experiments on Jin–Xin, Chaplygin gas, and two-phase flow demonstrate AP behavior across stiff, intermediate, and non-stiff regimes. The work provides robust, unified AP schemes applicable to multiscale hyperbolic-relaxation problems with practical relevance to fluid and kinetic-type systems.

Abstract

This paper presents the construction of two numerical schemes for the solution of hyperbolic systems with relaxation source terms. The methods are built by considering the relaxation system as a whole, without separating the resolution of the convective part from that of the source term. The first scheme combines the centered FORCE approach of Toro and co-authors with the unsplit strategy proposed by B{é}reux and Sainsaulieu. The second scheme consists of an approximate Riemann solver which carefully handles the source term approximation. The two schemes are built to be asymptotic preserving, in the sense that their limit schemes are consistent with the equilibrium model as the relaxation parameter tends to zero, without any CFL restriction. For specific models, it is possible to prove that they preserve invariant domains and admit a discrete entropy inequality.

Asymptotic preserving schemes for hyperbolic systems with relaxation

TL;DR

The paper addresses numerical approximation of hyperbolic systems with relaxation by introducing two asymptotic-preserving schemes that solve the full relaxation system without splitting the convective and source terms. The first scheme embeds the centered FORCE flux within an unsplit staggered framework, while the second uses an approximate Riemann solver that accounts for the source term, both designed to converge to the equilibrium model as and to the fully non-relaxed hyperbolic system as . For specific models, invariant-domain preservation and a discrete entropy inequality are established (notably for Jin–Xin), and extensive numerical experiments on Jin–Xin, Chaplygin gas, and two-phase flow demonstrate AP behavior across stiff, intermediate, and non-stiff regimes. The work provides robust, unified AP schemes applicable to multiscale hyperbolic-relaxation problems with practical relevance to fluid and kinetic-type systems.

Abstract

This paper presents the construction of two numerical schemes for the solution of hyperbolic systems with relaxation source terms. The methods are built by considering the relaxation system as a whole, without separating the resolution of the convective part from that of the source term. The first scheme combines the centered FORCE approach of Toro and co-authors with the unsplit strategy proposed by B{é}reux and Sainsaulieu. The second scheme consists of an approximate Riemann solver which carefully handles the source term approximation. The two schemes are built to be asymptotic preserving, in the sense that their limit schemes are consistent with the equilibrium model as the relaxation parameter tends to zero, without any CFL restriction. For specific models, it is possible to prove that they preserve invariant domains and admit a discrete entropy inequality.

Paper Structure

This paper contains 18 sections, 5 theorems, 94 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Continuous setting
  3. The general case
  4. Some exemples
  5. The Jin and Xin model
  6. Chaplygin gas model
  7. Two-phase flow model
  8. Staggered scheme
  9. Definition of the scheme
  10. Properties of the staggered scheme
  11. An approximate Riemann solver accounting for the source term
  12. Definition of the scheme
  13. Properties of the Approximate Riemann solver
  14. Numerical comparison of the two schemes
  15. The Jin and Xin model
  16. ...and 3 more sections

Key Result

Proposition 1

Let the constant sequence of cell-averaged values $(\mathbf{W}_j^{(1),n},\mathbf{W}_j^{(2),n})$ be given at time $t^n$, for $j\in \mathbb Z$. Under the CFL condition CFLstaggered, the scheme eq:sch-final is asymptotic preserving, in the sense that it is consistent with solutions of the hyperbolic mo

Figures (12)

  • Figure 1:
  • Figure 2: Juxtaposition of the approximate Riemann solvers defining the sequence $\mathbf W_j^{n+1}$, $j\in \mathbb Z$, using the intermediate states $\mathbf W_{j\pm\frac{1}{2}}^*$.
  • Figure 3: Solutions for the limiting behavior $\varepsilon=10^{-6}$ of Jin-Xin model on a 500-cell mesh at $t_{final}=3.2$
  • Figure 4: Comparison of the solutions obtained with the numerical schemes on a 500-cell mesh at final time $T = 0.1$ for $\varepsilon=1$ (top) and $\varepsilon=40$ (bottom).
  • Figure 5: $L^2$-norm error between the numerical solution and an exact solution to the Burger's equation for small values of $\varepsilon$ at fixed $\Delta x=4 \times 10^{-3}$ (top) and $\Delta x=2 \times 10^{-3}$ (bottom).
  • ...and 7 more figures

Theorems & Definitions (11)

  • Proposition 1: Asymptotic preserving property
  • proof
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • Remark 1
  • Proposition 3
  • Proposition 4
  • proof
  • ...and 1 more