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Monotone (A,B) entropy stable numerical scheme for ScalarConservation 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 inspace, 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. Forsolutions corresponding to a connection (A,B), there exists convergent numerical schemesbased 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., usedwidely 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 fromthe singular mapping technique of Temple and chain estimate of Adimurthi and Gowda, weprove 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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References

Adimurthi, and Veerappa Gowda, G.D., Conservation laws with discontinuous flux. J. Math. Kyoto Univ. 43 (2003) 27–70. Google Scholar
Adimurthi, , Dutta, R., Ghoshal, Shyam Sundar and Veerappa Gowda, G.D., Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux. Commun. Pure Appl. Math. 64 (2011) 84–115. Google Scholar
Adimurthi, , Jaffré, J. and Veerappa Gowda, G.D., Godunov type methods for scalar conservation laws with flux function discontinuous in the space variable. SIAM J. Numer. Anal. 42 (2004) 179–208. Google Scholar
Adimurthi, , Mishra, S. and Veerappa Gowda, G.D., Explicit Hopf-Lax type formulas for Hamilton-Jacobi equations and conservation laws with discontinuous coefficients. J. Differ. Equ. 241 (2007) 1–31. Google Scholar
Adimurthi, , Mishra, S. and Veerappa Gowda, G.D., Optimal entropy solutions for conservation laws with discontinuous flux-functions. J. Hyperbolic Differ. Equ. 2 (2005) 783–837. Google Scholar
Andreianov, B., Karlsen, K.H. and Risebro, N.H., A theory of L1-dissipative solvers for scalar conservation laws with discontinuous flux. Arch. Ration. Mech. Anal. 201 (2011) 27–86. Google Scholar
Bürger, R., Karlsen, K.H., Risebro, N.H. and Towers, J.D., Well-posedness in BVt and convergence of a difference scheme for continuous sedimentation in ideal clarifier thickener units. Numer. Math. 97 (2004) 25–65. Google Scholar
Bürger, R., Karlsen, K.H., Risebro, N.H. and Towers, J. D., Monotone difference approximations for the simulation of clarifier-thickener units. Comput. Vis. Sci. 6 (2004) 83–91. Google Scholar
Bürger, R., Karlsen, K.H. and Towers, J. D., An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections. SIAM J. Numer. Anal. 47 (2009) 1684–1712. Google Scholar
Crandall, , Michael, G. and Andrew, Majda, Monotone difference approximations for scalar conservation laws. Math. Comput. 34 (1980) 1–21. Google Scholar
S. Diehl, Conservation Laws with Applications to Continuous Sedimentation, Doctoral Dissertation. Lund University, Lund, Sweden (1995). Google Scholar
Diehl, S., A conservation laws with point source and discontinuous flux function modelling continuous sedimentation. SIAM J. Appl. Math. 56 (1996) 388–419. Google Scholar
T. Gimse and N.H. Risebro, Riemann problems with discontinuous flux function, Proc. of 3rd Internat. Conf. Hyperbolic Problems, Studentlitteratur, Uppsala (1991) 488–502. Google Scholar
Jérôme, Jaffré and Mishra, S., On the upstream mobility scheme for two-phase flow in porous media. Comput. Geosci. 14 (2010) 105–124 Google Scholar
Kaasschieter, E., Solving the Buckley-Leverret equation with gravity in a heterogeneous porous media. Comput. Geosci. 3 (1999) 23–48. Google Scholar
Karlsen, K.H. and Towers, J.D., 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. Google Scholar
Klingenberg, C. and Risebro, N.H., Convex conservation laws with discontinuous coefficients, existence, uniqueness and asymptotic behavior. Commun. Partial Differ. Equ. 20 (1995) 1959–1990. Google Scholar
Keyfitz, B., Solutions with shocks: An example of an L 1-contractive semi-group. Commun. Pure Appl. Math. 24 (1971) 125–132. Google Scholar
S. Mishra, Analysis and Numerical approximation of conservation laws with discontinuous coefficients, Ph.D. thesis, Indian Institute of Science, Bangalore (2005). Google Scholar
Mochon, S., An analysis for the traffic on highways with changing surface conditions. Math. Model. 9 (1987) 1–11. Google Scholar
Nessyahu, H. and Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408–463. Google Scholar
Seguin, N. and Vovelle, J., Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients. Math. Models Methods Appl. Sci. 13 (2003) 221–257. Google Scholar
Temple, B. and Isaacson, E., Nonlinear resonance in systems of conservation laws. SIAM J. Appl. Math. 52 (1992) 1260–1278. Google Scholar
Towers, J.D., A difference scheme for conservation laws with a discontinuous flux: the nonconvex case. SIAM J. Numer. Anal. 39 (2001) 1197–1218. Google Scholar
Towers, J.D., Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Numer. Anal. 38 (2000) 681–698. Google Scholar
S. Tveit, Numerical methods for hyperbolic conservation laws with discontinuous flux. Master of Science Thesis in Reservoir Mechanics, University of Bergen (2011). Google Scholar