Preprint 2003-077

Convergence of Upwind Finite Difference Algorithms for Scalar Conservation Laws with Indefinite Discontinuities in the Flux Function

Siddhartha Mishra

Abstract: We consider the following scalar conservation law with flux function discontinuous in the space variable i.e,

ut + (H(x)f(u)+((1-H(x))g(u))x =0
where H is the Heaviside function and f and g are smooth under the assumptions that either f is convex (concave) and g is concave (convex). We show existence of a weak solution by showing finite difference schemes of the Godunov and Enquist Osher type converge to a weak solution. Uniqueness follows from a Kruzkhov type argument. We also give explcit solutions of the Riemann problem. At the level of numerics, we give easy to implement numerical schemes of the Godunov and Enquist Osher type and report numerical experiments comparing our schemes with other existing schemes. The central feature of this paper is the modification of the singular mapping technique (the main analytical tool for equations of this type) which allows us to prove convergence for the schemes. Equations of the above type occour when we consider scalar conservation laws with sign changing discontinuous coefficients.

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Siddhartha Mishra, <>
Publishing information:
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<> December 22 2003.

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