Fast accurate computation of the fully nonlinear solitary surface gravity waves

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.

💡 Research Summary

The paper presents a compact yet powerful algorithm for computing steady solitary gravity waves—solutions of the free‑surface Euler equations in irrotational flow. The authors begin by applying a conformal mapping that transforms the physical domain with a deformable free surface into a fixed rectangular domain in the complex plane. This mapping preserves the Laplace equation governing the velocity potential while converting the nonlinear free‑surface boundary conditions into equivalent conditions on a static computational grid. Consequently, the free‑surface elevation η(x) becomes the sole unknown function defined on a fixed interval.

Next, the transformed governing equations and boundary conditions are combined to eliminate the velocity potential, yielding a single nonlinear integro‑differential equation for η. Unlike traditional formulations that involve a coupled system of potential and surface variables, this reduction dramatically lowers the dimensionality of the problem. The nonlinear term appears as a square‑root expression, which is advantageous for the iterative scheme that follows.

To solve the reduced equation, the authors adopt Petviashvili’s iteration method. This fixed‑point‑type algorithm is known for its super‑linear convergence when applied to equations possessing a dominant linear operator and a homogeneous nonlinear term. In each iteration the nonlinear part is scaled by a carefully chosen factor that enforces convergence, while the linear part is solved efficiently in Fourier space.

The spatial discretisation is performed with a pseudo‑spectral approach. The surface elevation η(x) is expanded in a periodic Fourier series, and all derivatives are evaluated via the Fast Fourier Transform (FFT). Because the FFT dominates the computational cost, the overall algorithm scales as O(N log N), where N is the number of spectral modes. The iteration typically converges within 10–20 steps to machine precision (relative errors below 10⁻¹⁴) for grid sizes ranging from 2¹⁰ to 2¹⁴ points. The authors also demonstrate that, when combined with arbitrary‑precision arithmetic libraries, the method can achieve any prescribed number of significant digits, effectively eliminating round‑off limitations.

Extensive numerical experiments compare the new scheme with conventional finite‑difference and boundary‑element methods. The results show that the present algorithm is orders of magnitude faster while delivering superior accuracy, even for highly nonlinear waves with large amplitudes. Parameter studies reveal robust convergence across a wide range of wave heights and speeds, confirming that the method remains stable in regimes where other solvers struggle.

Finally, the paper discusses the broader applicability of the technique. Because the core ideas—conformal mapping, reduction to a single surface equation, Petviashvili iteration, and FFT‑based pseudo‑spectral discretisation—are not tied to the specific 2‑D irrotational setting, they can be extended to multi‑layer fluids, weakly rotational flows, and even three‑dimensional wave problems with appropriate modifications. The authors conclude that the algorithm provides a highly efficient, scalable, and arbitrarily accurate tool for researchers and engineers working on nonlinear water‑wave phenomena, wave‑energy conversion devices, and related coastal‑engineering applications.