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An instability of the Godunov scheme

TitleAn instability of the Godunov scheme
Publication TypeJournal Article
Year of Publication2006
AuthorsBressan, A, Jenssen, HK, Baiti, P
JournalComm. Pure Appl. Math. 59 (2006) 1604-1638

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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