Monotone $\left(A,B\right)$ entropy stable numerical scheme for Scalar Conservation Laws with discontinuous flux
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 6, p. 1725-1755

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.

DOI : https://doi.org/10.1051/m2an/2014017
Classification:  35L45,  35L60,  35L65,  35L67
Keywords: conservation laws, discontinuous flux, Lax−Friedrichs scheme, singular mapping, interface entropy condition, (A,b)connection
@article{M2AN_2014__48_6_1725_0,
author = {Adimurthi and Dutta, Rajib and Veerappa Gowda, G. D. and Jaffr\'e, J\'er\^ome},
title = {Monotone $(A,B)$ entropy stable numerical scheme for Scalar Conservation Laws with discontinuous flux},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {48},
number = {6},
year = {2014},
pages = {1725-1755},
doi = {10.1051/m2an/2014017},
language = {en},
url = {http://www.numdam.org/item/M2AN_2014__48_6_1725_0}
}

Adimurthi; Dutta, Rajib; Veerappa Gowda, G. D.; Jaffré, Jérôme. Monotone $(A,B)$ entropy stable numerical scheme for Scalar Conservation Laws with discontinuous flux. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 6, pp. 1725-1755. doi : 10.1051/m2an/2014017. http://www.numdam.org/item/M2AN_2014__48_6_1725_0/

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