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.