Thomas Anderson (Caltech)

Title: Hybrid frequency/time methods for efficient and parallel high-order solution of wave scattering problems

Abstract: Hybrid frequency/time numerical methods for time-dependent problems use synthesis of frequency-domain solutions to produce causal transient solutions (e.g. to the wave equation). A well-studied and popular member of this class is the convolution quadrature method, which is at its core based on time-stepping and discrete Laplace transformation. We develop and propose a novel alternative method using as a model problem that of acoustic sound-soft scattering in two- and three-dimensional space (the ideas are fully generalizable). Relying on Fourier transformation in time (with a new time-windowing strategy for incident waves and efficient and accelerated oscillatory quadrature techniques for transform inversion), the method utilizes a fixed (time-independent) number of frequency-domain integral-equation solutions to evaluate time domain solutions for arbitrarily long times with no dispersion error and high-order accuracy. The algorithm enables parallelization in time in a straightforward manner and it allows for time leaping---that is, solution sampling at any given time T at O(1)-bounded sampling cost. The proposed strategy for time-domain problems provides significant advantages compared with alternatives such as volumetric discretization, integral equation time-stepping, and convolution-quadrature approaches.

Bio: Thomas Anderson is a Ph.D. Candidate in Applied & Computational Mathematics at the California Institute of Technology, in the Computing + Mathematical Sciences department. Before that he completed his B.S. in Applied Mathematics at the New Jersey Institute of Technology.