Ahmad Abassi

UC Berkeley

“Semi-Analytic Numerical Methods for Standing Water Waves in Finite Depth”

We generalize the semi-analytic theory of standing water waves from infinite to finite depth. We use high-performance computing to numerically solve the problem and study the analytical properties of the solution.


We build on and generalize the pioneering works of Schwartz and Whitney, and Amick and Toland to create and develop a semi-analytic theory of standing water waves in finite depth. We start by proposing an ansatz, appropriate to the finite-depth setting and consistent with the infinite-depth setting, for the Fourier expansion of the solution functions, as well as Stokes-type expansions for the temporal Fourier coefficients in terms of the wave amplitude. We then derive the system of ordinary differential equations governing our problem and propose an iterative algorithm to demonstrate the existence and uniqueness of a solution in the space of trigonometric polynomials, concluding with numerical experiments using high-performance computing in a hybrid C++ MPI/OpenMP environment with arbitrary-precision arithmetic to further investigate the analytically-intractable problem. In order to develop a computationally-feasible framework on modern supercomputers, we apply Bell polynomials from combinatorics to allow for the computation of the Stokes-type coefficients of the hyperbolic functions that arise in the finite-depth setting in quadratic time.

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