An Instability of the Godunov Scheme

TL;DR

The paper demonstrates that the fully discrete Godunov scheme for a specially crafted strictly hyperbolic 2×2 system can produce arbitrarily large downstream oscillations due to resonance between a moving shock and the grid. By constructing explicit inviscid and discrete traveling-wave solutions and analyzing the total variation of the discrete solution, the authors show that no general a priori BV bounds or $L^1$ stability estimates hold for such finite-difference schemes. They establish that, although convergence to entropy solutions may still be possible, the standard BV-based approaches fail for fully discrete schemes. The work highlights intrinsic limitations of discrete methods for hyperbolic systems and underscores the need for alternative analytical frameworks, such as compensated compactness, in establishing convergence properties. It also provides a detailed mechanism via a discrete Cole–Hopf transform and heat-kernel estimates to understand grid-resonant oscillations.

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.

Figures (3)

• Figure 1: Discrete sources at integer points.
• Figure 2: Interaction of $G_x$ and fractional parts.
• Figure 3: Oscillations produced by sources along the curve $\gamma(t)$.

Theorems & Definitions (18)

• Lemma 3.1
• proof
• Lemma 4.1
• proof
• Lemma 5.1
• proof
• Lemma 5.2
• proof
• Lemma 5.3
• proof