A numerical scheme for singular shock solutions and a study of its consistence in the sense of distributions

In this paper we present a numerical scheme for the approximation of singular shock solutions of the Keyfitz-Kranzer model system. Consistence in the sense of distributions is studied. As long as some numerical properties are verified when the space step tends to 0, we prove that the scheme provides a numerical solution that satisfies the equations in the sense of distributions with an approximation that tends to 0 when h \rightarrow 0. We also show that this scheme adapts to degenerate systems. This is illustrated by two examples: the system presenting delta wave solutions originally studied by Korchinski and another system studied by Keyfitz-Kranzer that models elasticity. Consistence of the scheme in the sense of distributions is fully proved in the case of the Korchinski model.