Numerical methods for nonlinear interactions between water waves
Olmez, H. S.
Milgram, J. H.
Journal of Computational Physics
Ölmez, H. S., & Milgram, J. H. (1995). Numerical methods for nonlinear interactions between water waves. Journal of Computational Physics, 118(1), 62-72.
Please cite this item using this persistent URLhttp://hdl.handle.net/11693/10763
Two numerical methods for studying nonlinear interactions between spatially periodic water waves of disparate length scales are explored. Both use a time-stepping procedure, but one solves the boundary value problem at each time step by a boundary integral equation and the other uses a high-order spectral method. The central issues are the restrictions on accuracy of the spectral method associated with its inherent internal perturbation expansion and the propagation of numerical boundary integral method errors from the edge of the computational domain into its interior. The spectral method was found to be accurate for larger values of the product of long-wave amplitude and short-wave number than one might expect. Two methods of correcting numerical errors at each time step in the boundary integral equation method are explored. One evaluates the vertical derivative of the velocity potential at the surface by use of computed values of the potential on the vertical sides of the domain. The other replaces computed values by interpolated values based on spatial periodicity. Using either of these allows the boundary integral method to be used for larger values of the product of long-wave amplitude and short-wave number and for steeper waves than can be handled by the spectral method, but at substantially greater computational expense.
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