Structural stability of finite dispersion-relation preserving schemes

TL;DR

This paper investigates the stability of finite dispersion-relation preserving schemes, revealing that they can produce long-lived spurious solitary waves that cause persistent numerical errors, indicating structural instability.

Contribution

It extends previous analysis of classical schemes to dispersion-relation preserving schemes, identifying conditions under which spurious solitary waves occur and affect stability.

Findings

Spurious solitary waves can occur in finite-difference solutions.

Such waves can have very long lifetimes, causing persistent errors.

The study extends previous work to dispersion-relation preserving schemes.

Abstract

The goal of this work is to determine classes of travelling solitary wave solutions for a differential approximation of a finite difference scheme by means of a hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurance of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to has a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solution of the original continuous equations. This paper extends our previous work about classical schemes to dispersion-relation preserving schemes.