A Stabilised Nodal Spectral Element Method for Fully Nonlinear Water Waves

TL;DR

This paper introduces a high-order spectral element method for simulating fully nonlinear water waves, addressing aliasing instability issues with stabilization techniques, and demonstrating high accuracy and efficiency through benchmark tests.

Contribution

The paper develops a stabilized, high-order spectral element method for nonlinear water waves, incorporating over-integration and spectral filtering to ensure stability and accuracy.

Findings

Achieves exponential convergence in benchmark tests

Effectively stabilizes nonlinear terms with minimal computational overhead

Demonstrates high accuracy and speedup over traditional methods

Abstract

We present an arbitrary-order spectral element method for general-purpose simulation of non-overturning water waves, described by fully nonlinear potential theory. The method can be viewed as a high-order extension of the classical finite element method proposed by Cai et al (1998) \cite{CaiEtAl1998}, although the numerical implementation differs greatly. Features of the proposed spectral element method include: nodal Lagrange basis functions, a general quadrature-free approach and gradient recovery using global $L^2$ projections. The quartic nonlinear terms present in the Zakharov form of the free surface conditions can cause severe aliasing problems and consequently numerical instability for marginally resolved or very steep waves. We show how the scheme can be stabilised through a combination of over-integration of the Galerkin projections and a mild spectral filtering on a per element basis. This effectively removes any aliasing driven instabilities while retaining the high-order accuracy of the numerical scheme. The additional computational cost of the over-integration is found insignificant compared to the cost of solving the Laplace problem. The model is applied to several benchmark cases in two dimensions. The results confirm the high order accuracy of the model (exponential convergence), and demonstrate the potential for accuracy and speedup. The results of numerical experiments are in excellent agreement with both analytical and experimental results for strongly nonlinear and irregular dispersive wave propagation. The benefit of using a high-order -- possibly adapted -- spatial discretization for accurate water wave propagation over long times and distances is particularly attractive for marine hydrodynamics applications.