CMX Lunch Seminar: Thomas Anderson

"Fast hybrid" frequency/time techniques for efficient and parallelizable high-order transient wave scattering simulation

We propose a frequency/time hybrid integral-equation (though other frequency-domain methods are readily usable) method for the time-dependent wave equation in two and three-dimensional spatial domains. Relying on Fourier transformation in time, the method utilizes a fixed (time-independent) number of frequency-domain integral-equation solutions to evaluate time domain solutions for arbitrarily long times. The approach relies on two main elements, namely, 1) A smooth time-windowing methodology that enables accurate band-limited representations for arbitrary long time signals, and 2) A novel Fourier transform approach which, in a time-parallel manner and without causing spurious periodicity effects, delivers numerically-dispersionless spectrally-accurate solutions. The algorithm can tackle complex physical structures, it 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 frequency/time hybridization strategy, which generalizes to any linear partial differential equation in the time domain for which frequency-domain solutions can be obtained (including e.g. the time-domain Maxwell equations or time domain problems posed with dispersive media), provides significant advantages over other available alternatives such as volumetric discretization and convolution-quadrature approaches.