Overdetermined Shooting Methods for Computing Standing Water Waves with   Spectral Accuracy
  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 counter-propagating 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. 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. 
Jon Wilkening and Jia Yu
physics.flu-dyn