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A Wave Front Tracking Scheme for Flux Reconstruction in $2\times 2$ Hyperbolic Conservation Laws

Chaohua Duan, Yan Jiang, Hongyu Liu, Wenjian Peng

TL;DR

This work addresses the inverse problem of identifying flux functions in coupled $2\times2$ hyperbolic conservation laws from fixed-time Riemann observations. It develops a wave-front tracking framework that uses an equivalent shock concept to replace rarefaction fans and constructs piecewise quadratic $C^1$ flux surrogates $f_{1,m},f_{2,m}$ on grids with spacings $\delta$ and $\eta$, achieving provable convergence and stability: $\|f_1-f_{1,m}\|_\infty \le L_1\eta^2$, $\|f_2-f_{2,m}\|_\infty \le L_2\delta^2$, and $\|f_1'-f_{1,m}'\|_\infty,\|f_2'-f_{2,m}'\|_\infty \le 3L_1\eta,3L_2\delta$, with solution stability $\|S_T^{F_m}u^m - S_T^{F}u_{obs}\|_1 \le C T (L_1\eta + L_2\delta)$. Under $C^3$ regularity, convergence improves to cubic for function values and quadratic for derivatives, yielding $\|S_T^{F_m}u^m - S_T^{F}u_{obs}\|_1 = O(T(\eta^2+\delta^2))$. The framework is demonstrated on the isentropic Euler equations and the p-system, enabling extraction of the equation of state $p(\rho)$ from dynamic measurements. This provides a rigorous, practically implementable path to a fundamental inverse problem in continuum mechanics with potential impact on experimental and computational fluid dynamics.

Abstract

This paper introduces a novel wave front tracking framework for reconstructing unknown flux functions in $2\times 2$ hyperbolic conservation laws, extending beyond the well-studied scalar case. By analyzing Riemann solutions at fixed observation times, we develop explicit reconstruction formulas that handle arbitrary combinations of shock and rarefaction waves through a unified equivalent shock concept. Our method constructs piecewise quadratic $C^1$ flux approximations with rigorous convergence guarantees: the approximation errors decrease quadratically with the discretization parameters for function values and linearly for derivatives under $C^{1,1}$ regularity, with enhanced cubic and quadratic convergence respectively under $C^3$ regularity. Applications to the isentropic Euler equations and the mathematically equivalent p-system in compressible fluid dynamics demonstrate the method's capability to identify complete equations of state from limited dynamic measurements, providing a systematic approach to a fundamental inverse problem in continuum mechanics.

A Wave Front Tracking Scheme for Flux Reconstruction in $2\times 2$ Hyperbolic Conservation Laws

TL;DR

This work addresses the inverse problem of identifying flux functions in coupled hyperbolic conservation laws from fixed-time Riemann observations. It develops a wave-front tracking framework that uses an equivalent shock concept to replace rarefaction fans and constructs piecewise quadratic flux surrogates on grids with spacings and , achieving provable convergence and stability: , , and , with solution stability . Under regularity, convergence improves to cubic for function values and quadratic for derivatives, yielding . The framework is demonstrated on the isentropic Euler equations and the p-system, enabling extraction of the equation of state from dynamic measurements. This provides a rigorous, practically implementable path to a fundamental inverse problem in continuum mechanics with potential impact on experimental and computational fluid dynamics.

Abstract

This paper introduces a novel wave front tracking framework for reconstructing unknown flux functions in hyperbolic conservation laws, extending beyond the well-studied scalar case. By analyzing Riemann solutions at fixed observation times, we develop explicit reconstruction formulas that handle arbitrary combinations of shock and rarefaction waves through a unified equivalent shock concept. Our method constructs piecewise quadratic flux approximations with rigorous convergence guarantees: the approximation errors decrease quadratically with the discretization parameters for function values and linearly for derivatives under regularity, with enhanced cubic and quadratic convergence respectively under regularity. Applications to the isentropic Euler equations and the mathematically equivalent p-system in compressible fluid dynamics demonstrate the method's capability to identify complete equations of state from limited dynamic measurements, providing a systematic approach to a fundamental inverse problem in continuum mechanics.

Paper Structure

This paper contains 11 sections, 2 theorems, 104 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Technical developments and discussion
  4. Preliminaries and Main Results
  5. Mathematical Framework
  6. Main Results
  7. Proofs of Main Results
  8. Wave Front Tracking Analysis
  9. Flux Reconstruction and Convergence Analysis
  10. Enhanced Convergence under Higher Regularity
  11. Application to Isentropic Euler Equations and the p-System

Key Result

Theorem 2.3

Let $T>0$ and let $\Omega \subset \mathbb{R}^2$ be an open domain containing the rectangle $\left[u_*, u^*\right] \times \left[v_*, v^*\right]$, where $u_*<u^*$ and $v_*<v^*$. Let $c_1, c_2 \in \mathbb{R}$ be fixed. Assume that the flux functions $f_1, f_2: \mathbb{R} \rightarrow \mathbb{R}$ are $C^ where $L_1$ and $L_2$ are the Lipschitz constants of $f_1'$ and $f_2'$ on the respective intervals.

Figures (7)

  • Figure 1: Second-family shock wave (front shock)
  • Figure 2: Second-family rarefaction wave (front rarefaction)
  • Figure 3: Wave pattern classification: Regions I-IV
  • Figure 4: Two shock waves connecting states $(u_h,v_h)$ and $(u_{h+1},v_{h+1})$ through intermediate state $(u_{h+\frac{1}{2}},v_{h+\frac{1}{2}})$ for system \ref{['Main:eqf']}.
  • Figure 5: First-family shock wave followed by second-family rarefaction wave for system \ref{['Main:eqf']}.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Definition 2.1: Standard Riemann Semigroup
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 3.1
  • Remark 4.1: Lagrangian Formulation and the p-System
  • Theorem 4.2