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Well-balanced schemes for conservation laws with source terms based on a local discontinuous flux formulation

Authors: Kenneth Hvistendahl Karlsen, Siddhartha Mishra and Nils Henrik Risebro
Journal: Math. Comp. 78 (2009), 55-78
MSC (2000): Primary 35L65, 74S10, 65M12
Published electronically: September 17, 2008
MathSciNet review: 2448697
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Abstract: We propose and analyze a finite volume scheme of the Godunov type for conservation laws with source terms that preserve discrete steady states. The scheme works in the resonant regime as well as for problems with discontinuous flux. Moreover, an additional modification of the scheme is not required to resolve transients, and solutions of nonlinear algebraic equations are not involved. Our well-balanced scheme is based on modifying the flux function locally to account for the source term and to use a numerical scheme especially designed for conservation laws with discontinuous flux. Due to the difficulty of obtaining $ BV$ estimates, we use the compensated compactness method to prove that the scheme converges to the unique entropy solution as the discretization parameter tends to zero. We include numerical experiments in order to show the features of the scheme and how it compares with a well-balanced scheme from the literature.

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Additional Information

Kenneth Hvistendahl Karlsen
Affiliation: Centre of Mathematics for Applications (CMA), University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway

Siddhartha Mishra
Affiliation: Centre of Mathematics for Applications (CMA) , University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway

Nils Henrik Risebro
Affiliation: Centre of Mathematics for Applications (CMA), University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway

Keywords: Conservation law, discontinuous solution, source term, finite volume scheme, well-balanced scheme, convergence, compensated compactness method
Received by editor(s): November 9, 2006
Received by editor(s) in revised form: October 10, 2007
Published electronically: September 17, 2008
Additional Notes: The third author was supported in part by an Outstanding Young Investigators Award from the Research Council of Norway.
Article copyright: © Copyright 2008 American Mathematical Society

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