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Asymptotic-preserving schemes for the initial-boundary value problem of hyperbolic relaxation systems

Yizhou Zhou

TL;DR

The paper addresses stable and accurate simulation of IBVPs for hyperbolic relaxation systems in the stiff limit $\epsilon\to0$. It develops asymptotic-preserving (AP) schemes that solve the relaxation system on coarse meshes while capturing equilibrium behavior and boundary layers, extending AP concepts from IVP to IBVPs and applying them to interface problems. A boundary-AP discretization based on $M(\eta)=A^{-1}(I-\eta Q)$ adaptively handles stiffness, while a generalized Kreiss condition framework ensures well-posed limits and correct boundary data. Numerical experiments with Jin-Xin models and linear relaxation systems demonstrate robust boundary/interface resolution and first-order accuracy on coarse meshes, validating the practical impact of the unified AP approach across both IBVPs and interface problems.

Abstract

In this work, we present a numerical method for the initial-boundary value problem (IBVP) of first-order hyperbolic systems with source terms. The scheme directly solves the relaxation system using a relatively coarse mesh and captures the equilibrium behavior quite well, even in the presence of boundary layers. This method extends the concept of asymptotic-preserving schemes from initial-value problems to IBVPs. Moreover, we apply this idea to design a unified numerical scheme for the interface problem of relaxation systems.

Asymptotic-preserving schemes for the initial-boundary value problem of hyperbolic relaxation systems

TL;DR

The paper addresses stable and accurate simulation of IBVPs for hyperbolic relaxation systems in the stiff limit . It develops asymptotic-preserving (AP) schemes that solve the relaxation system on coarse meshes while capturing equilibrium behavior and boundary layers, extending AP concepts from IVP to IBVPs and applying them to interface problems. A boundary-AP discretization based on adaptively handles stiffness, while a generalized Kreiss condition framework ensures well-posed limits and correct boundary data. Numerical experiments with Jin-Xin models and linear relaxation systems demonstrate robust boundary/interface resolution and first-order accuracy on coarse meshes, validating the practical impact of the unified AP approach across both IBVPs and interface problems.

Abstract

In this work, we present a numerical method for the initial-boundary value problem (IBVP) of first-order hyperbolic systems with source terms. The scheme directly solves the relaxation system using a relatively coarse mesh and captures the equilibrium behavior quite well, even in the presence of boundary layers. This method extends the concept of asymptotic-preserving schemes from initial-value problems to IBVPs. Moreover, we apply this idea to design a unified numerical scheme for the interface problem of relaxation systems.

Paper Structure

This paper contains 11 sections, 2 theorems, 100 equations, 5 figures.

Table of Contents

  1. Introduction
  2. An illustrative example: Jin-Xin model
  3. Initial-boundary value problem
  4. Classical first-order upind IMEX scheme
  5. Boundary Asymptotic-preserving treatment
  6. General relaxation system
  7. Relaxation limits for IBVPs of relaxation systems
  8. Boundary AP treatment
  9. Interface problem
  10. Numerical experiment
  11. Missing Proof

Key Result

Lemma 2.2

\newlabellem2.10 For sufficiently small $\eta$, it follows that where $L^A_+$ and $L^A_-$ are left-eigenvectors of $A$ associated with positive and negative eigenvalues. On the other hand, for sufficiently large $\eta$, we have

Figures (5)

  • Figure 1: $L^2$, $L^1$ and $L^\infty$ errors between the numerical solutions and reference solutions for different cases: (left) the stiff case with boundary-layer; (middle) the stiff case without boundary-layer; (right) the non-stiff case.
  • Figure 2: (Left) the numerical results at $t=0.2$ with boundary-AP treatment; (Right) the numerical results at $t=0.2$ by upwind scheme.
  • Figure 3: The numerical results at $t=0.3$ by our boundary-AP method and by the classical upwind method.
  • Figure 4: (left) the numerical results by the upwind scheme with $N_x=100$; (middle) the results by our boundary-AP method with $N_x=100$; (right) the results by the upwind scheme with $N_x=1000$.
  • Figure 5: The numerical results by our boundary-AP method (left) and by the classical upwind method (right).

Theorems & Definitions (7)

  • Remark 2.1
  • Lemma 2.2
  • Remark 2.3
  • Definition 3.1: Generalized Kreiss condition MR1722195
  • Definition 3.2: Right-stable/unstable matrix MR1722195
  • Lemma 3.3
  • Proof 1: Proof of Lemma \ref{['lem2.1']}