Formal proof of flux conservation for Roe solver on isothermal Euler (ρ, ρu)

Prove that the Roe inter-cell flux for the isothermal Euler equations, restricted to the density component ρ and x-momentum component ρu, satisfies the flux jump condition F(U_R) − F(U_L) = A(U_L, U_R)[U_R − U_L], where F(U) is the isothermal Euler flux vector and A(U_L, U_R) is the Roe matrix consistent with the flux Jacobian in the limit U_L, U_R → U, thereby establishing exact conservation (jump continuity) of ρ and ρu across cell interfaces.

Background

The paper develops a formal verification pipeline for first-order hyperbolic PDE solvers, including Lax–Friedrichs and Roe schemes, and proves a variety of correctness properties (hyperbolicity preservation, stability, Lipschitz continuity, and flux conservation) across several model systems. For the isothermal Euler system, the authors factorize the equations and attempt to prove properties for each coupled pair of conserved components.

While the framework successfully proves many properties for linear advection, inviscid Burgers’, and perfectly hyperbolic Maxwell’s equations, it encounters difficulties on the isothermal Euler system. The authors explicitly note that the only property for which their theorem-prover truly fails—despite being unconditionally true—is the flux conservation (jump continuity) condition for the Roe solver applied to the ρ and ρu components. This identifies a concrete open verification task within their pipeline.