Convergence of Lax–Friedrichs and Godunov schemes for a nonstrictly hyperbolic system of conservation laws arising in oil recovery

This paper is devoted to the compactness framework and the convergence theorem for the Lax–Friedrichs and Godunov schemes applied to a 2 × 2 system of non-strictly hyperbolic nonlinear conservation laws that arises from mathematical models for oil recovery. The presence of a degeneracy in the hyperbolicity of the system requires a careful analysis of the entropy functions, whose regularity is necessary to obtain the result. For this purpose, it is necessary to combine the classical techniques referring to a singular Euler–Poisson–Darboux equation with the compensated compactness method.