# New conformal mapping for adaptive resolving of the complex singularities of Stokes wave

## TL;DR

This paper introduces a new conformal mapping technique that improves the numerical computation of Stokes waves by effectively managing complex singularities, enabling detailed analysis of waves approaching their limiting form.

## Contribution

A novel conformal mapping method that accelerates convergence in computing Stokes waves and allows detailed exploration of their limiting behavior.

## Key findings

- Enhanced numerical convergence for Stokes wave computations.
- Successful resolution of waves near the limiting wave of greatest height.
- Generalization of the mapping to handle multiple singularities.

## Abstract

A new highly efficient method is developed for computation of traveling periodic waves (Stokes waves) on the free surface of deep water. A convergence of numerical approximation is determined by the complex singularites above the free surface for the analytical continuation of the travelling wave into the complex plane. An auxiliary conformal mapping is introduced which moves singularities away from the free surface thus dramatically speeding up numerical convergence by adapting the numerical grid for resolving singularities while being consistent with the fluid dynamics. The efficiency of that conformal mapping is demonstrated for Stokes wave approaching the limiting Stokes wave (the wave of the greatest height) which significantly expands the family of numerically accessible solutions. It allows to provide a detailed study of the oscillatory approach of these solutions to the limiting wave. Generalizations of the conformal mapping to resolve multiple singularities are also introduced.