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

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 convolu...

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Bibliographic Details
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
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Summary: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.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2010.08.025