Overdetermined shooting methods for computing standing water waves with spectral accuracy

Abstract

A high-performance shooting algorithm is developed to compute time-periodic solutions of the free-surface Euler equations with spectral accuracy in double and quadruple precision. The method is used to study resonance and its effect on standing water waves. We identify new nucleation mechanisms in which isolated large-amplitude solutions, and closed loops of such solutions, suddenly exist for depths below a critical threshold. We also study degenerate and secondary bifurcations related to Wilton's ripples in the traveling case, and explore the breakdown of self-similarity at the crests of extreme standing waves. In shallow water, we find that standing waves take the form of counterpropagating solitary waves that repeatedly collide quasi-elastically. In deep water with surface tension, we find that standing waves resemble counter-propagating depression waves. We also discuss existence and non-uniqueness of solutions, and smooth versus erratic dependence of Fourier modes on wave amplitude and fluid depth.In the numerical method, robustness is achieved by posing the problem as an overdetermined nonlinear system and using either adjoint-based minimization techniques or a quadratically convergent trust-region method to minimize the objective function. Efficiency is achieved in the trust-region approach by parallelizing the Jacobian computation so the setup cost of computing the Dirichlet-to-Neumann operator in the variational equation is not repeated for each column. Updates of the Jacobian are also delayed until the previous Jacobian ceases to be useful. Accuracy is maintained using spectral collocation with optional mesh refinement in space, a high order Runge-Kutta or spectral deferred correction method in time, and quadruple-precision for improved navigation of delicate regions of parameter space as well as validation of double-precision results. Implementation issues for GPU acceleration are briefly discussed, and the performance of the algorithm is tested for a number of hardware configurations.

Computation of three-dimensional standing water waves

“…The four mixed mode rays would correspond to a Wilton's ripple phenomenon [49,50,51,52,53] in which multiple wavelengths are present in the leading order asymptotics. Although this is speculation, it is consistent with recent finding in the two-dimensional case [33], where it was computationally feasible to resolve the degenerate bifurcation and study how it splits under perturbation of the fluid depth. It is also consistent with Bridges' analysis [22] of degenerate bifurcations for three-dimensional water waves in the weakly nonlinear regime.…”

“…The minimization procedure is carried out in the overdetermined shooting framework of Wilkening and Yu [33], which has been found to be more robust than a standard shooting method in which the number of nonlinear equations is equal to the number of unknowns. The problem is overdetermined because the initial conditions are zero-padded in Fourier space while the residual vector r contains ϕ at all grid points.…”

“…Details of how the trust region radius ∆ should be varied when the Jacobian updates are delayed over several residual computations are given in Ref. [33].…”

“…For comparison, the thin dashed black overlays are calculated using the boundary integral method of Ref. [33] for deep water, which has very similar dynamics to the current regime. Since the functions f r have a wavelength of 2π / √ 2, it is necessary to rescale the boundary integral results by 1 / √ 2 to make the comparison.…”

Liquid sloshing in a horizontally forced vessel with bottom topography

“…For large amplitude waves and waves with sharp interfaces, the conformal mapping technique is less effective (due to surface grid points being poorly distributed at the sharp crests) and in these cases, methods such as the boundary element method are more effective (e.g. Wilkening and Yu (2012)). …”