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

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.