Convergence of implicit Finite Volume methods for scalar conservation laws with discontinuous flux function

¹ Université Paris-Sud XI / Laboratoire de Mathématiques - CNRS UMR 8628, Bât. 425, 91405 Orsay Cedex, France.
² ENS Cachan Antenne de Bretagne / IRMAR - CNRS UMR 6625, Av. R. Schuman, Campus de Ker Lann, 35170 Bruz, France. Julien.Vovelle@bretagne.ens-cachan.fr

Received: 24 May 2007
Revised: 4 December 2007
Revised: 28 February 2008

Abstract

This paper deals with the problem of numerical approximation in the Cauchy-Dirichlet problem for a scalar conservation law with a flux function having finitely many discontinuities. The well-posedness of this problem was proved by Carrillo [J. Evol. Eq. 3 (2003) 687–705]. Classical numerical methods do not allow us to compute a numerical solution (due to the lack of regularity of the flux). Therefore, we propose an implicit Finite Volume method based on an equivalent formulation of the initial problem. We show the well-posedness of the scheme and the convergence of the numerical solution to the entropy solution of the continuous problem. Numerical simulations are presented in the framework of Riemann problems related to discontinuous transport equation, discontinuous Burgers equation, discontinuous LWR equation and discontinuous non-autonomous Buckley-Leverett equation (lubrication theory).