High performance numerics for water waves

sws Jon Wilkening gave a talk on high performance numerics for computing 2D and 3D time-periodic standing waves.  The numerical methods are spectral in space with 8-15th order time integration.  Two strategies were used to reduce the problem to the free surface: a dirichlet-neumann operator reduction using a double-layer potential, and in 2D infinite depth a time-dependent conformal mapping was used.  Main result in 2D is that no highest wave of the Penney-Price type was found; instead a complex branching structure with a sequence of high waves, as seen in the above figure, but no limiting wave.  A class of quasi-periodic standing waves was also computed, by interpolating between pure standing waves (zero momentum) and travelling waves (maximum momentum for fixed energy).  Quadruple precision was used to confirm the accuracy of the results. A range of results for 3D was also presented, including standing waves in vessels with square cross section. A video of the talk is available here.

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