Computation of three-dimensional standing gravity-capillary waves in deep water
Xin Guan (Imperial)
In this talk, we present a numerical investigation of three-dimensional standing gravity-capillary waves in deep water. We employ high-order truncated models, derived from the Hamiltonian formulation of water waves and the Craig–Sulem expansion of the Dirichlet–Neumann operator, to approximate the original full potential problem. This approach provides an explicit, computationally efficient approximation of the global relation between the normal velocity of the water surface and its velocity potential. We validate the effectiveness and accuracy of the models by comparing the numerical solutions with existing analytic and numerical results. To find exact time periodic solutions, we directly solve a boundary-value problem which is triply periodic in spatial and temporal directions, rather than embedding initial-value solvers into a Newton or a nonlinear least-squares solver. This eliminates the CFL constraint on the time step and avoids the numerical instabilities related to time marching, which are particularly severe in the presence of surface tension. To lower the computational cost, we exploit the spatial-temporal symmetries of three kinds of standing waves and reduce the number of unknowns to either 1/16 or 1/32 of the original, according to the intrinsic symmetries of solutions. Using this numerical method, we investigate the nonlinear gravity-capillary standing waves bifurcating from linear solutions and find that typical large-amplitude solutions feature rounded crests, in contrast to the sharp pyramidal-shape crests of pure gravity standing waves. Due to nonlinear resonances, bifurcation may break into disjoint branches in adjacent to specific frequencies, similar to those occurring for two-dimensional standing waves. When this happens, particular high-order temporal harmonics become as significant as the primary one, leading to complex surface patterns. We investigate standing waves close to the linearly resonant regime, and find that solutions are in the form of a combination of the two resonant components. Direct temporal evolutions confirm the excellent numerical accuracy of the computed standing waves, and illustrate the cyclic recurrence when resonances happen.
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