Accelerated Cartesian expansion (ACE) based framework for the rapid evaluation of diffusion, lossy wave, and Klein–Gordon potentials

• Published in: Journal of computational physics 2010-12, Vol.229 (24), p.9119-9134
• Main Authors: Vikram, M., Baczewski, A., Shanker, B., Kempel, L.
• Format: Article
• Language: English
• Subjects: Accelerated Cartesian expansion (ACE); Algorithms; Block-Toeplitz; Cartesian; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Computational techniques; Convolution; Diffusion; Exact sciences and technology; Fast multipole methods; Lossy wave; Mathematical analysis; Mathematical methods in physics; Mathematical models; Physics; Temporal logic; Transient
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2010.08.025

Diffusion, lossy wave, and Klein–Gordon equations find numerous applications in practical problems across a range of diverse disciplines. The temporal dependence of all three Green’s functions are characterized by an infinite tail. This implies that the cost complexity of the spatio-temporal convolutions, associated with evaluating the potentials, scales as O N s 2 N t 2 , where N s and N t are the number of spatial and temporal degrees of freedom, respectively. In this paper, we discuss two new methods to rapidly evaluate these spatio-temporal convolutions by exploiting their block-Toeplitz nature within the framework of accelerated Cartesian expansions (ACE). The first scheme identifies a convolution relation in time amongst ACE harmonics and the fast Fourier transform (FFT) is used for efficient evaluation of these convolutions. The second method exploits the rank deficiency of the ACE translation operators with respect to time and develops a recursive numerical compression scheme for the efficient representation and evaluation of temporal convolutions. It is shown that the cost of both methods scales as O ( N s N t log 2 N t ) . Several numerical results are presented for the diffusion equation to validate the accuracy and efficacy of the fast algorithms developed here.