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Convergence of Godunov type methods for a conservation law with a spatially varying discontinuous flux function

Authors: Adimurthi, Siddhartha Mishra and G. D. Veerappa Gowda
Journal: Math. Comp. 76 (2007), 1219-1242
MSC (2000): Primary 35L65, 65M06, 65M12
Published electronically: January 25, 2007
MathSciNet review: 2299772
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Abstract: We deal with single conservation laws with a spatially varying and possibly discontinuous coefficient. This equation includes as a special case single conservation laws with conservative and possibly singular source terms. We extend the framework of optimal entropy solutions for these classes of equations based on a two-step approach. In the first step, an interface connection vector is used to define infinite classes of entropy solutions. We show that each of these classes of solutions is stable in $L^1$. This allows for the possibility of choosing one of these classes of solutions based on the physics of the problem. In the second step, we define optimal entropy solutions based on the solution of a certain optimization problem at the discontinuities of the coefficient. This method leads to optimal entropy solutions that are consistent with physically observed solutions in two-phase flows in heterogeneous porous media. Another central aim of this paper is to develop suitable numerical schemes for these equations. We develop and analyze a set of Godunov type finite volume methods that are based on exact solutions of the corresponding Riemann problem. Numerical experiments are shown comparing the performance of these schemes on a set of test problems.

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Additional Information

Affiliation: TIFR center, P.O. Box 1234, Bangalore 560012, India

Siddhartha Mishra
Affiliation: Center of Mathematics for Applications, University of Oslo, P.O. Box 1053, Oslo–0316, Norway

G. D. Veerappa Gowda
Affiliation: TIFR center, P.O.Box 1234, Bangalore 560012, India

Received by editor(s): September 19, 2005
Received by editor(s) in revised form: June 23, 2006
Published electronically: January 25, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.