Hamiltonian Finite Element Discretization for Nonlinear Free Surface Water Waves

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-dep...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 73; no. 1; pp. 366 - 394
Main Authors: Brink, Freekjan, Izsák, Ferenc, van der Vegt, J. J. W.
Format: Journal Article
Language:English
Published: New York Springer US 01.10.2017
Springer Nature B.V
Subjects:
Accuracy
Algorithms
Amplitudes
Approximation
Computational Mathematics and Numerical Analysis
Decades
Discretization
Free surfaces
Hamiltonian functions
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Numerical analysis
Potential flow
Surface water
Theoretical
Time dependence
Time integration
Velocity
Water waves
Wave equations
Wave interaction
Finite element method
Hamiltonian systems
65P10
Nonlinear potential flow water wave equations
76B15
Symplectic time integration
65M60
Moving meshes
ISSN:0885-7474, 1573-7691
Online Access:Get full text
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Summary: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.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-017-0416-9