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 shocks. As long as some properties (such as boundedness of the velocity) are verified when the space step hh tends to 0, we prove that the scheme provides approximate solutions that tend to satisfy the equations. More precisely, when the approximate solutions are plugged into the equations the result tends to 0 when h→0h→0 in the familiar definition of weak solutions, with the requirement of a smooth test function. These properties can be fully proved for general classes of systems extending the Korchinski system, that do not have distribution solutions in the usual sense. In other cases, such as the Keyfitz–Kranzer singular shocks, these properties have been checked up to very small values of hh. These results explain numerical observations on very different systems.