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.

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/

[1] Adimurthi and G.D. Veerappa Gowda, Conservation laws with discontinuous flux. J. Math. Kyoto Univ. 43 (2003) 27-70. | MR 2028700 | Zbl 1063.35114

[2] Dutta, Shyam Sundar Ghoshal and G.D. Veerappa Gowda, Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux. Commun. Pure Appl. Math. 64 (2011) 84-115. | MR 2743877 | Zbl 1223.35222

[3] Jaffré and G.D. Veerappa Gowda, Godunov type methods for scalar conservation laws with flux function discontinuous in the space variable. SIAM J. Numer. Anal. 42 (2004) 179-208. | MR 2051062 | Zbl 1081.65082

[4] Mishra and G.D. Veerappa Gowda, Explicit Hopf-Lax type formulas for Hamilton-Jacobi equations and conservation laws with discontinuous coefficients. J. Differ. Equ. 241 (2007) 1-31. | MR 2356208 | Zbl 1128.35067

[5] Mishra and G.D. Veerappa Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions. J. Hyperbolic Differ. Equ. 2 (2005) 783-837. | MR 2195983 | Zbl 1093.35045

[6] A theory of L1-dissipative solvers for scalar conservation laws with discontinuous flux. Arch. Ration. Mech. Anal. 201 (2011) 27-86. | MR 2807133 | Zbl 1261.35088

, and ,[7] Well-posedness in BVt and convergence of a difference scheme for continuous sedimentation in ideal clarifier thickener units. Numer. Math. 97 (2004) 25-65. | MR 2045458 | Zbl 1053.76047

, , and ,[8] Monotone difference approximations for the simulation of clarifier-thickener units. Comput. Vis. Sci. 6 (2004) 83-91. | MR 2061269 | Zbl 1299.76283

, , and ,[9] An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections. SIAM J. Numer. Anal. 47 (2009) 1684-1712. | MR 2505870 | Zbl 1201.35022

, and ,[10] Michael and Majda, Andrew, Monotone difference approximations for scalar conservation laws. Math. Comput. 34 (1980) 1-21. | MR 551288 | Zbl 0423.65052

[11] Conservation Laws with Applications to Continuous Sedimentation, Doctoral Dissertation. Lund University, Lund, Sweden (1995). | MR 2714835

,[12] A conservation laws with point source and discontinuous flux function modelling continuous sedimentation. SIAM J. Appl. Math. 56 (1996) 388-419. | MR 1381652 | Zbl 0849.35142

,[13] Riemann problems with discontinuous flux function, Proc. of 3rd Internat. Conf. Hyperbolic Problems, Studentlitteratur, Uppsala (1991) 488-502. | MR 1109304 | Zbl 0789.35102

and ,[14] Jaffré, Jérôme and S. Mishra, On the upstream mobility scheme for two-phase flow in porous media. Comput. Geosci. 14 (2010) 105-124 | MR 2579833 | Zbl pre05675960

[15] Solving the Buckley-Leverret equation with gravity in a heterogeneous porous media. Comput. Geosci. 3 (1999) 23-48. | MR 1696184 | Zbl 0952.76085

,[16] Convergence of the Lax−Friedrichs scheme and stability for conservation laws with a discontinuous space-time dependent flux. Chinese Ann. Math. Ser. B 25 (2004) 287-318. | MR 2086124 | Zbl 1112.65085

and ,[17] Convex conservation laws with discontinuous coefficients, existence, uniqueness and asymptotic behavior. Commun. Partial Differ. Equ. 20 (1995) 1959-1990. | MR 1361727 | Zbl 0836.35090

and ,[18] Solutions with shocks: An example of an L1-contractive semi-group. Commun. Pure Appl. Math. 24 (1971) 125-132. | MR 271545 | Zbl 0209.12401

,[19] Analysis and Numerical approximation of conservation laws with discontinuous coefficients, Ph.D. thesis, Indian Institute of Science, Bangalore (2005).

,[20] An analysis for the traffic on highways with changing surface conditions. Math. Model. 9 (1987) 1-11. | MR 898784

,[21] Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408-463. | MR 1047564 | Zbl 0697.65068

and ,[22] Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients. Math. Models Methods Appl. Sci. 13 (2003) 221-257. | MR 1961002 | Zbl 1078.35011

and ,[23] Nonlinear resonance in systems of conservation laws. SIAM J. Appl. Math. 52 (1992) 1260-1278. | MR 1182123 | Zbl 0794.35100

and ,[24] A difference scheme for conservation laws with a discontinuous flux: the nonconvex case. SIAM J. Numer. Anal. 39 (2001) 1197-1218. | MR 1870839 | Zbl 1055.65104

,[25] Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Numer. Anal. 38 (2000) 681-698. | MR 1770068 | Zbl 0972.65060

,[26] Numerical methods for hyperbolic conservation laws with discontinuous flux. Master of Science Thesis in Reservoir Mechanics, University of Bergen (2011).

,