Journal article · Preprint article
A stabilised nodal spectral element method for fully nonlinear water waves
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) [5], 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 L2 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 discretisation for accurate water wave propagation over long times and distances is particularly attractive for marine hydrodynamics applications.
Language: | English |
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Publisher: | Elsevier BV |
Year: | 2016 |
Pages: | 1-21 |
ISSN: | 10902716 and 00219991 |
Types: | Journal article and Preprint article |
DOI: | 10.1016/j.jcp.2016.04.060 |
ORCIDs: | Engsig-Karup, Allan Peter and 0000-0003-3504-7530 |
Finite element methods High-order discretisation Hydrodynamics Nonlinear and dispersive free surface waves SDG 14 - Life Below Water Spectral element method Unstructured mesh