An Instability of the Godunov Scheme

TL;DR

This paper demonstrates that the Godunov scheme can generate unbounded oscillations in solutions to hyperbolic conservation laws due to resonance effects, challenging assumptions about stability and bounded variation in numerical schemes.

Contribution

It provides a counterexample showing the instability of the Godunov scheme and the failure of BV bounds and $L^1$ stability for finite difference methods in certain hyperbolic systems.

Findings

Godunov scheme can produce arbitrarily large oscillations

Resonance occurs when shock speed is close to rational

No general BV bounds or $L^1$ stability for finite difference schemes

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.