Revisiting the method of characteristics via a convex hull algorithm

We revisit the method of characteristics for shock wave solutions to nonlinear hyperbolic problems and we describe a novel numerical algorithm —the convex hull algorithm (CHA) — in order to compute, both, entropy dissipative solutions (satisfying all relevant entropy inequalities) and entropy conservative (or mul-tivalued) solutions to nonlinear hyperbolic conservation laws. Our method also applies to Hamilton-Jacobi equations and other problems endowed with a method of characteristics. From the multivalued solutions determined by the method of characteristic, our algorithm ”extracts ” the entropy dissipative solutions, even after the formation of shocks. It applies to, both, convex or non-convex flux/Hamiltonians. We demonstrate the relevance of the proposed approach with a variety of numerical tests including a problem from fluid dynamics