Abstract
A novel finite element discretization for nonlinear potential flow water waves is presented. Starting from Luke’s Lagrangian formulation we prove that an appropriate finite element discretization preserves the Hamiltonian structure of the potential flow water wave equations, even on general time-dependent, deforming and unstructured meshes. For the time-integration we use a modified Störmer–Verlet method, since the Hamiltonian system is non-autonomous due to boundary surfaces with a prescribed motion, such as a wave maker. This results in a stable and accurate numerical discretization, even for large amplitude nonlinear water waves. The numerical algorithm is tested on various wave problems, including a comparison with experiments containing wave interactions resulting in a large amplitude splash.
Highlights
The numerical simulation of nonlinear water waves is a challenging problem
The potential flow water wave equations, when expressed in terms of the free surface potential and wave height, have a Hamiltonian structure, but this structure is generally lost in the numerical discretization
This will directly result in an energy preserving numerical discretization that is stable for large amplitude nonlinear waves
Summary
The numerical simulation of nonlinear water waves is a challenging problem. These waves appear naturally in the ocean, rivers and lakes and greatly affect the motion of ships and induce significant forces on floating and fixed structures. The main topic of this article is to develop a finite element discretization for which we can explicitly prove that it preserves the Hamiltonian structure, even on time-dependent meshes that are needed to follow the free surface motion. This will directly result in an energy preserving numerical discretization that is stable for large amplitude nonlinear waves. These methods, typically require evaluating a singular integration kernel and tend to require the evaluation of dense matrix–vector products, which have to be solved with a fast multipole method to keep the computational complexity approximately linear These methods do not automatically preserve the Hamiltonian structure of the potential flow water wave equations.
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