Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I

We propose a new numerical approach to computing nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keeps nonclassical shock waves as sharp interfaces, unlike standard finite difference schemes. The main challenge is to achieve, at the discretization level, a consistency property with respect to a prescribed kinetic relation. The latter is required for the selection of physically meaningful nonclassical shocks. Our method is based on a reconstruction technique performed in each computational cell that may contain a nonclassical shock. To validate this approach, we establish several consistency and stability properties, and we perform careful numerical experiments. The convergence of the algorithm toward the physically meaningful solutions selected by a kinetic relation is demonstrated numerically for several test cases, including concave-convex as well as convex-concave flux functions.