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

Karlsen, Kenneth; Mishra, Siddhartha; Risebro, Nils

doi:10.1090/S0025-5718-08-02117-0

Well-balanced schemes for conservation laws with source terms based on a local discontinuous flux formulation
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by Kenneth Hvistendahl Karlsen, Siddhartha Mishra and Nils Henrik Risebro;

Math. Comp. 78 (2009), 55-78

DOI: https://doi.org/10.1090/S0025-5718-08-02117-0

Published electronically: September 17, 2008

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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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