ABSTRACT We study the convergence and consistency of the Godunov–Roe method for hyperbolic systems of balance laws with source terms depending on a piecewise constant function. By introducing appropriate families of paths, we define weak solutions in the nonconservative framework and analyze their relation to Roe‐type schemes. We establish uniform bounds on the total variation of the numerical solutions under the C.F.L and monotonicity conditions. Furthermore, we prove convergence of the Roe scheme to a weak solution in the sense of Dal Maso–LeFloch–Murat.
Vinh et al. (2026) studied this question.