An instability of the Godunov scheme

Alberto Bressan; Helge Kristian Jenssen; Paolo Baiti

doi:10.1002/cpa.20141

An instability of the Godunov scheme

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Abstract

We construct a solution to a 2 × 2 strictly hyperbolic system of conservation laws, showing that the Godunov scheme [13] 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.

Original language	English (US)
Pages (from-to)	1604-1638
Number of pages	35
Journal	Communications on Pure and Applied Mathematics
Volume	59
Issue number	11
DOIs	https://doi.org/10.1002/cpa.20141
State	Published - Nov 2006

All Science Journal Classification (ASJC) codes

General Mathematics
Applied Mathematics

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10.1002/cpa.20141

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title = "An instability of the Godunov scheme",

abstract = "We construct a solution to a 2 × 2 strictly hyperbolic system of conservation laws, showing that the Godunov scheme [13] 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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AU - Baiti, Paolo

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N2 - We construct a solution to a 2 × 2 strictly hyperbolic system of conservation laws, showing that the Godunov scheme [13] 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.

AB - We construct a solution to a 2 × 2 strictly hyperbolic system of conservation laws, showing that the Godunov scheme [13] 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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JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

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An instability of the Godunov scheme

Abstract

All Science Journal Classification (ASJC) codes

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