A stabilised nodal spectral element method for fully nonlinear water waves

Autor: Daniele Bigoni, Claes Eskilsson, Allan Peter Engsig-Karup
Rok vydání: 2016
Předmět:
FOS: Computer and information sciences
Physics and Astronomy (miscellaneous)
Wave propagation
G.1
Spectral element method
FOS: Physical sciences
Basis function
Geometry
01 natural sciences
010305 fluids & plasmas
Computational Engineering
Finance
and Science (cs.CE)
0103 physical sciences
FOS: Mathematics
Mathematics - Numerical Analysis
0101 mathematics
Computer Science - Computational Engineering
Finance
and Science
Galerkin method
Mathematics
Numerical Analysis
Applied Mathematics
Mathematical analysis
Numerical Analysis (math.NA)
Mixed finite element method
Computational Physics (physics.comp-ph)
Finite element method
Computer Science Applications
010101 applied mathematics
Computational Mathematics
Nonlinear system
Modeling and Simulation
Physics - Computational Physics
Numerical stability
Zdroj: Journal of Computational Physics. 318:1-21
ISSN: 0021-9991
DOI: 10.1016/j.jcp.2016.04.060
Popis: 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.
Accepted for publication in Journal of Computational Physics April 29, 2016
Databáze: OpenAIRE
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