Fast Time Domain Integral Equation Solvers for Analyzing Two-Dimensional Scattering Phenomena; Part I: Temporal Acceleration

Abstract

The primary impediment to analyzing two-dimensional transient scattering phenomena using classical marching-on-in-time–based integral equation solvers is these schemes' high computational complexity that scales as O (Ns 2 Nt 2, where Ns and Nt denote the number of spatial and temporal degrees of freedom of the current on the scatterer. Here, three schemes that reduce this cost by permitting the rapid evaluation of the temporal convolution of a bandlimited transient source signature with the two-dimensional wave equation Green function are studied; these three methods rely on (i) blocked fast Fourier transforms, (ii) truncated singular value decompositions, and (iii) multiresolution concepts. The computational cost of all three proposed algorithms scales as O(Ns 2Nt logα Nt) with α ≤ 2. The three schemes are compared on their respective multiplicative constants inherent in this cost estimate, their memory requirements, and their ease of implementation.