Preprint 2005-009

An Instability of the Godunov Scheme

Alberto Bressan, Helge Kristian Jenssen and Paolo Baiti

Abstract: We construct a solution to a 22 strictly hyperbolic system of conservation laws, showing that the Godunov scheme \cite{Godunov59} can produce an arbitrarily large amount of oscillations. This happens when the speed of a shock is close to rational, inducing a resonance with the grid. Differently from the Glimm scheme or the vanishing viscosity method, for systems of conservation laws our counterexample indicates that no a priori BV bounds or L1 stability estimates can in general be valid for finite difference schemes.

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Alberto Bressan, <>
Helge Kristian Jenssen, <>
Paolo Baiti, <>
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<> February 1 2005.

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