On the Rate of Error Growth in Time for Numerical Solutions of Nonlinear Dispersive Wave Equations

TL;DR

This paper investigates how numerical errors in solving nonlinear dispersive wave equations grow over time, showing that energy-conserving methods tend to have slower error growth compared to non-conserving ones.

Contribution

It extends previous findings by providing numerical evidence that error growth behavior applies broadly across various equations and numerical methods.

Findings

Error grows quadratically in non-conservative methods

Error grows linearly in conservative methods

Results are supported by extensive numerical experiments

Abstract

We study the numerical error in solitary wave solutions of nonlinear dispersive wave equations. A number of existing results for discretizations of solitary wave solutions of particular equations indicate that the error grows quadratically in time for numerical methods that do not conserve energy, but grows only linearly for conservative methods. We provide numerical experiments suggesting that this result extends to a very broad class of equations and numerical methods.