Long-term behaviour of symmetric partitioned linear multistep methods II. Invariants error analysis for some nonlinear dispersive wave models

TL;DR

This paper analyzes the long-term behavior of symmetric partitioned linear multistep methods when used to numerically solve nonlinear dispersive wave equations, focusing on invariants and error analysis for solitary wave solutions.

Contribution

It extends previous theory to nonlinear dispersive PDEs, providing invariants error analysis for PLMMs applied to these models with numerical validation.

Findings

PLMMs effectively approximate solitary wave solutions

Error analysis reveals invariants preservation over long time

Numerical experiments support theoretical predictions

Abstract

In this paper, the use of partitioned linear multistep methods (PLMM) as time integrators for the numerical approximation of some partial differential equations (pdes) is studied. We consider the periodic initial-value problem of two nonlinear dispersive wave models as case studies. From the spatial discretization with pseudospectral methods, the theory developed for PLMMs by the authors in a previous companion paper is applied to analyze the time integration with PLMMs of the semidiscrete equations when approximating solitary wave solutions. The results are illustrated with some numerical experiments. In addition, a computational study is performed in an exploratory fashion to analyze the extension of the results to the approximation of more general localized solutions.