Monotone (A,B) entropy stable numerical scheme for Scalar Conservation Laws with discontinuous flux

¹ TIFR-CAM, PB 6503, Sharadanagar, 560065 Bangalore, India.
aditi@math.tifrbng.res.in; rajib@math.tifrbng.res.in; gowda@math.tifrbng.res.in
² INRIA, BP 105, 78153 Le Chesnay Cedex, France.
jerome.jaffre@inria.fr

Received: 8 February 2013
Revised: 2 September 2013

Abstract

For scalar conservation laws in one space dimension with a flux function discontinuous in space, there exist infinitely many classes of solutions which are L¹ contractive. Each class is characterized by a connection (A,B) which determines the interface entropy. For solutions corresponding to a connection (A,B), there exists convergent numerical schemes based on Godunov or Engquist−Osher schemes. The natural question is how to obtain schemes, corresponding to computationally less expensive monotone schemes like Lax−Friedrichs etc., used widely in applications. In this paper we completely answer this question for more general (A,B) stable monotone schemes using a novel construction of interface flux function. Then from the singular mapping technique of Temple and chain estimate of Adimurthi and Gowda, we prove the convergence of the schemes.