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

TL;DR

This paper develops a wave front tracking method for reconstructing flux functions in 2x2 hyperbolic conservation laws, enabling accurate flux approximation and equation of state identification from limited measurements.

Contribution

It introduces a unified wave front tracking framework with explicit reconstruction formulas for flux functions, extending scalar methods to systems with rigorous convergence guarantees.

Findings

Quadratic error decay in flux approximation

Successful application to Euler equations and p-system

Ability to identify complete equations of state from limited data

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.