Ahmad Abassi

UC Berkeley

“The Semi-Analytic Theory and Computation of Wilton Ripples”

We aim to study and numerically simulate Wilton ripples, interesting mathematical and physical phenomena, to gain insights into their analytical properties. We begin by proposing a conformal map and an appropriate ansatz for its Fourier series to pull back the physical domain of the problem into conformal space, followed by an analysis of the regular (non-resonant) problem where we propose appropriate Stokes expansions and derive an iterative algorithm to compute the Stokes coefficients using several perturbation-theoretic results. We then repeat the analysis and derive appropriate algorithms for the resonant problem with the appropriate Stokes expansions and prove the existence of formal series expansions of Wilton ripples by demonstrating using a computer that our algorithms do not break down.
Finally, we present the results of our computational study carried out in a parallel, hybrid MPI/OpenMP C++ environment using multiple-precision arithmetic to numerically compute the Stokes coefficients in the regular and resonant problems to empirically determine whether the Wilton ripples’ expansions have a positive radius of convergence and are therefore analytic.

ABSTRACT

We use semi-analytic methods to derive algorithms to compute amplitude-series expansions of Wilton ripples in conformal space. Our main goal is to numerically simulate our algorithms on a supercomputer to determine whether the series expansions of Wilton ripples have a positive, finite radius of convergence.
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