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Published April 27, 2020 | Published + Submitted
Journal Article Open

High-order, Dispersionless "Fast-Hybrid" Wave Equation Solver. Part I: O(1) Sampling Cost via Incident-Field Windowing and Recentering


This paper proposes a frequency/time hybrid integral-equation 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, with superalgebraically small errors, 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 arbitrarily 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. A similar hybrid technique can be obtained on the basis of Laplace transforms instead of Fourier transforms, but we do not consider the Laplace-based method in the present contribution. The algorithm can handle dispersive media, it 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, for arbitrarily large values of T, and without requirement of evaluation of the solution at intermediate times. 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) and which is applicable in a wide range of scientific and engineering contexts, provides significant advantages over other available alternatives, such as volumetric discretization, time-domain integral equations, and convolution quadrature approaches.

Additional Information

© 2020 Society for Industrial and Applied Mathematics. Submitted to the journal's Methods and Algorithms for Scientific Computing section March 25, 2019; accepted for publication (in revised form) January 27, 2020; published electronically April 27, 2020. This work was supported by the AFOSR through grant FA9550-15-1-0043, by the NSF through grant DMS-1714169, by DARPA through grant HR00111720035, and by the NSSEFF Vannevar Bush Fellowship under contract number N00014-16-1-2808. The work of the first author was also supported by the DOE through grant DEFG02-97ER25308. Thanks are due to Emmanuel Garza for facilitating the use of the existing 3D frequency-domain codes [20]. A number of valuable comments and suggestions by the reviewers are also thankfully acknowledged.

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Submitted - 1807.02718.pdf

Published - 19m1251953.pdf


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August 19, 2023
August 19, 2023