On Petviashvili type methods for traveling wave computations: Acceleration techniques

TL;DR

This paper enhances Petviashvili-type methods for computing traveling waves by integrating acceleration techniques, improving convergence, and broadening applicability in nonlinear dispersive wave equations through numerical experiments.

Contribution

It introduces acceleration techniques into fixed point algorithms for traveling wave computations, improving convergence and applicability.

Findings

Acceleration methods improve convergence in challenging cases

Vector extrapolation and Anderson acceleration are effective

Numerical experiments demonstrate enhanced performance

Abstract

In this paper a family of fixed point algorithms, generalizing the \PM method, is considered. A previous work studied the convergence of the methods. Presented here is a second part of the analysis, concerning the introduction of some acceleration techniques into the iterative procedures. The purpose of the research is two-fold: one is improving the performance of the methods in case of convergence and the second one is widening their application when generating traveling waves in nonlinear dispersive wave equations, transforming some divergent into convergent cases. Two families of acceleration techniques are considered: the vector extrapolation methods and the Anderson acceleration methods. A comparative study through several numerical experiments is carried out.