A convergent and conservative schemes for nonclassical solutions based on kinetic relations

AbstractWe propose a new numerical approach to compute nonclassical solutions to hyper-bolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces, contrary to 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 experi-ments. 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. Résume ́ Nous proposons un nouvel algorithme pour approcher les solutions non classiques de lois de conservation hyperboliques. Le schéma aux différences finies présente ́ ici est conser-vatif et transporte de manière exacte les chocs non classiques, a ̀ la différences des algorithmes standard. La principale difficulte ́ est de garantir, au niveau discret, la consistance avec une re