Fast accurate computation of the fully nonlinear solitary surface gravity waves

TL;DR

This paper introduces a fast, easy-to-implement algorithm for computing steady solitary gravity wave solutions with high accuracy, utilizing conformal mapping, Petviashvili's iterations, and FFT-based spectral methods.

Contribution

The paper presents a novel, efficient algorithm that simplifies and accelerates the computation of solitary gravity waves with arbitrary precision.

Findings

Algorithm achieves super-linear complexity with FFT

Results can be computed to arbitrary accuracy using multi-precision arithmetic

Method is easy to implement and faster than existing approaches

Abstract

In this short note, we present an easy to implement and fast algorithm for the computation of the steady solitary gravity wave solution of the free surface Euler equations in irrotational motion. First, the problem is reformulated in a fixed domain using the conformal mapping technique. Second, the problem is reduced to a single equation for the free surface. Third, this equation is solved using Petviashvili's iterations together with pseudo-spectral discretisation. This method has a super-linear complexity, since the most demanding operations can be performed using a FFT algorithm. Moreover, when this algorithm is combined with the multi-precision arithmetics, the results can be obtained to any arbitrary accuracy.