Full Euler Solver on Deep Water

Overview

The present collection of MATLAB scripts provides a high-fidelity Fourier pseudo-spectral solver for the full Euler equations with free surface on a fluid layer of infinite depth (the so-called deep water approximation). The time-dependent fluid domain is transformed into a fixed computational strip using the conformal mapping technique (pioneered by L.V. Ovsyannikov in 1972 and further developed by A. Dyachenko et al. in 1996).

Key Features

• Conformal Mapping: Transforms the time-dependent physical domain into a fixed computational strip
• Pseudo-spectral Method: High-order spatial discretization using FFT with exponential dealiasing
• Adaptive Time-stepping: Embedded Cash-Karp 5(4) method with PI control for optimal time step selection
• Integrating Factor Technique: Exact integration of linear dispersive terms
• Surface Velocity Computation: Optional post-processing to compute horizontal and vertical velocities (NEW)
• Conservation Monitoring: Real-time tracking of mass, momentum, and total energy

Mathematical Framework

The solver implements the following key mathematical components:

1. Spatial Discretization: Fourier pseudo-spectral method with N = 16384 modes
2. Time Integration: Cash-Karp embedded Runge-Kutta method of order 5(4)
3. Dealiasing: Exponential filter following Hou et al. with Λ = 15
4. Linear Terms: Exact integration using rotation matrices
5. Conformal Transform: x = ξ - H[η], where H is the Hilbert transform

Running the Simulation

To execute the simulation, simply run:

run_Euler

No command-line arguments are required as all parameters are defined within the script.

Configuration Options

The main script run_Euler.m includes a toggle for velocity computation:

compute_velocities = false;  % Set to true to compute surface velocities

When enabled, the solver computes:

• U: Horizontal surface velocity
• V: Vertical surface velocity
• U_x: Horizontal derivative of U
• V_x: Horizontal derivative of V

Code Structure

• run_Euler.m: Main simulation driver with adaptive time-stepping
• RHS.m: Computes nonlinear right-hand side for conformal Euler evolution
• turn.m: Handles linear rotation steps with spectral filtering
• Conf2Real.m: Transforms from conformal space to physical coordinates
• Plot.m: Real-time visualization of surface elevation and Fourier spectrum

Documentation

• doc/VelocityField.tex: Mathematical derivation of surface velocities and gradients in conformal mapping for water waves. This document provides:
  • Derivation of velocity components from the complex potential
  • Transformation formulas between physical and conformal coordinates
  • Laplace equation in conformal coordinates and the relationship between velocity potential and streamfunction
  • Explicit formulas for computing surface velocity gradients
  • Authors: Francesco Fedele (Georgia Tech) and Denys Dutykh (Khalifa University)

Physical Problem

The solver is initialized to simulate the celebrated Peregrine breather evolution in the full Euler equations. This is a fundamental nonlinear wave solution that models extreme wave events. We refer to the following publication for more details:

• L. Shemer & L. Alperovich. Peregrine breather revisited. Phys. Fluids, 25, 051701, 2013

Author

Denys Dutykh
Khalifa University of Science and Technology
Abu Dhabi, UAE
GitHub: https://github.com/dutykh/

Please don't hesitate to contact the author with any questions, remarks, and bug reports. The latest contact details can be found at:

• http://www.denys-dutykh.com

Acknowledgements

The author would like to thank the following colleagues (in alphabetical order) who helped develop and enhance this solver:

• Didier Clamond, University of Nice Sophia Antipolis, Laboratoire J.A. Dieudonné, France
  Contributions: Conformal mapping methodology and numerical implementation

• Bernard Ee, Tel Aviv University, School of Mechanical Engineering, Israel
  Contributions: Peregrine breather initialization and validation

• Francesco Fedele, Associate Professor, School of Civil & Environmental Engineering and School of Electrical & Computer Engineering, Georgia Institute of Technology, Atlanta, GA, USA
  Contributions: Surface velocity computation module and physical derivatives transformation. Special thanks for initiating and preparing the initial draft of the mathematical documentation (VelocityField.tex)

• Lev Shemer, Tel Aviv University, School of Mechanical Engineering, Israel
  Contributions: Peregrine breather theory and experimental validation

Version History

• v1.1 (2025): Added surface velocity computation capability with contributions from Prof. Francesco Fedele
• v1.0 (2025): Initial release with full Euler solver using conformal mapping technique

License

This software is distributed under the MIT License. See the LICENSE file for details.