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An Instability of the Godunov Scheme

Alberto Bressan·Helge Kristian Jenssen·Paolo Baiti
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Abstract

We construct a solution to a $2\times 2$ 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 $L^1$ stability estimates can in general be valid for finite difference schemes.

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Alberto Bressan, Helge Kristian Jenssen, Paolo Baiti. 2005-02-07. An Instability of the Godunov Scheme. https://arxiv.org/abs/math/0502125

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