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

Published online by Cambridge University Press:  26 September 2014

Adimurthi ,
Rajib Dutta ,
G. D. Veerappa Gowda  and
Jérôme Jaffré
Adimurthi
Affiliation:
TIFR-CAM, PB 6503, Sharadanagar, 560065 Bangalore, India.. aditi@math.tifrbng.res.in; rajib@math.tifrbng.res.in; gowda@math.tifrbng.res.in
Rajib Dutta
Affiliation:
TIFR-CAM, PB 6503, Sharadanagar, 560065 Bangalore, India.. aditi@math.tifrbng.res.in; rajib@math.tifrbng.res.in; gowda@math.tifrbng.res.in
G. D. Veerappa Gowda
Affiliation:
TIFR-CAM, PB 6503, Sharadanagar, 560065 Bangalore, India.. aditi@math.tifrbng.res.in; rajib@math.tifrbng.res.in; gowda@math.tifrbng.res.in
Jérôme Jaffré
Affiliation:
INRIA, BP 105, 78153 Le Chesnay Cedex, France.; jerome.jaffre@inria.fr
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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.

Keywords

Conservation lawsdiscontinuous fluxLax−Friedrichs schemesingular mappinginterface entropy condition(A,B)connection
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 6 , November 2014 , pp. 1725 - 1755
Copyright
© EDP Sciences, SMAI, 2014

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