New conformal mapping for adaptive resolving of the complex singularities of Stokes wave
Provides a more efficient numerical tool for fluid dynamics researchers studying extreme ocean waves.
Developed a new conformal mapping method that dramatically speeds up numerical convergence for computing Stokes waves near the limiting wave, enabling detailed study of the oscillatory approach to the wave of greatest height.
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.