JournalsifbVol. 10, No. 3pp. 399–421

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

  • Christophe Chalons

    Université Pierre et Marie Curie-Paris 6, France
  • Philippe G. LeFloch

    Université Pierre et Marie Curie - Paris 6, France
  • Benjamin Boutin

    Université Pierre et Marie Curie, Paris, France
  • Frédéric Lagoutière

    Université Pierre et Marie Curie, Paris, France
Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I cover
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Abstract

We propose a new numerical approach to compute nonclassical solutions to hyperbolic 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 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.

Cite this article

Christophe Chalons, Philippe G. LeFloch, Benjamin Boutin, Frédéric Lagoutière, Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I. Interfaces Free Bound. 10 (2008), no. 3, pp. 399–421

DOI 10.4171/IFB/195