We introduce a new stabilized high-order and unstructured numerical model for modeling fully nonlinear and dispersive water waves. The model is based on a nodal spectral element method of arbitrary order in space and a -transformed formulation due to Cai, Langtangen, Nielsen and Tveito (1998). In the present paper we use a single layer of quadratic (in 2D) and prismatic (in 3D) elements. The model has been stabilized through a combination of over-integration of the Galerkin projections and a mild modal filter. We present numerical tests of nonlinear waves serving as a proof-of-concept validation for this new high-order model. The model is shown to exhibit exponential convergence even for very steep waves and there is a good agreement to analytic and experimental data.
For accurate wave propagation the simpler alternative compared to solving the full three-dimensional Navier-Stokes equations, is to solve either the fully nonlinear and dispersive potential flow equations with an unsteady free surface, or subject to approximation hereof in the form of Boussinesq-type equations. Such models can account for most important wave phenomena such as diffraction, refraction, nonlinear wave-wave interactions and interaction with complex structures when the numerical scheme give support to do so. The most straightforward way is to use multi-element (multi-domain) numerical schemes with support for adaptive unstructured meshes. With regard to fully nonlinear potential flow (FNPF) models; the first use of classical low-order finite element methods for fully nonlinear water waves was proposed by Wu and Eatock Taylor (1994).
Most existing FNPF finite element models (Greaves, Wu, Borthwick and Eatock Taylor,1997; Ma, Wu and Eatock Taylor, 2001a; Ma, Wu and Eatock Taylor, 2001b; Wu and Eatock Taylor, 2003) use the Mixed Eulerian Lagrangian method (Longuet-Higgins and Cokelet, 1976) for updating the free surface variables. The mesh updating can be computationally expensive and alternative, cheaper ways have been proposed, e.g. the Quasi Arbitrary Lagrangian Eulerian finite element method (QALE-FEM) by Ma and Yan (2006).
