  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 multivalued) 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.
