  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.
