A compact higher-order finite-difference scheme for the wave equation can be strongly non-dissipative on non-uniform meshes

TL;DR

This paper investigates the stability of a high-order finite-difference scheme for the wave equation on non-uniform meshes, revealing conditions under which the scheme becomes non-dissipative and unstable, supported by numerical evidence.

Contribution

It establishes necessary stability conditions for a Numerov-type scheme on non-uniform meshes, highlighting potential exponential growth in solutions and practical limitations.

Findings

Uniform in time stability is impossible with complex eigenvalues.

The scheme can exhibit exponential growth, making it strongly non-dissipative.

Refined mesh sequences require strict step size conditions similar to explicit parabolic schemes.

Abstract

We study necessary conditions for stability of a Numerov-type compact higher-order finite-difference scheme for the 1D homogeneous wave equation in the case of non-uniform spatial meshes. We first show that the uniform in time stability cannot be valid in any spatial norm provided that the complex eigenvalues appear in the associated mesh eigenvalue problem. Moreover, we prove that then the solution norm grows exponentially in time making the scheme strongly non-dissipative and therefore impractical. Numerical results confirm this conclusion. In addition, for some sequences of refining spatial meshes, an excessively strong condition between steps in time and space is necessary (even for the non-uniform in time stability) which is familiar for explicit schemes in the parabolic case.