Preprint 2000-009

On the Convergence of Godunov Scheme for Nonlinear Hyperbolic Systems

Alberto Bressan and Helge Kristian Jenssen

Abstract: We consider systems of the form
ut+A(u)ux=0,     u \in Rn
where the matrix $A(u)$ is assumed to be strictly hyperbolic and with the property that the integral curves of the eigenvector fields are straight lines. For this class of systems one can define a natural Riemann solver, and hence a Godunov scheme, which generalize the standard Riemann solver and Godunov scheme for conservative systems. We show convergence and L$^1$ stability for this scheme when applied to data with small total variation. The main step in the proof is to estimate the increase in the total variation produced by the scheme due to quadratic coupling terms. Using Duhamel's principle, the problem is reduced to the estimate of the product of two Green kernels, representing probability densities of discrete random walks. The total amount of coupling is then determined by the expected number of crossings between two random walks with strictly different average speeds. This provides a discrete analogue of the arguments developed in \cite{Bianchini99, BressanShen99} in connection with continuous random processes.

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Alberto Bressan, <>
Helge Kristian Jenssen, <>
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<> February 21 2000.

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