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

## Abstract

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}). Furthermore, several numerical results are presented for the diffusion equation to validate the accuracy and efficacy of the fast algorithms developed here.

- Authors:

- Michigan State Univ., East Lansing, MI (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1252698

- Report Number(s):
- SAND-2016-1056J

Journal ID: ISSN 0021-9991; 619148

- Grant/Contract Number:
- AC04-94AL85000

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 229; Journal Issue: 24; Journal ID: ISSN 0021-9991

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; accelerated Cartesian expansion (ACE); diffusion; lossy wave; transient; Block-Toeplitz; fast multipole methods

### Citation Formats

```
Baczewski, Andrew David, Vikram, Melapudi, Shanker, Balasubramaniam, and Kempel, Leo. Accelerated Cartesian expansion (ACE) based framework for the rapid evaluation of diffusion, lossy wave, and Klein-Gordon potentials. United States: N. p., 2010.
Web. doi:10.1016/j.jcp.2010.08.025.
```

```
Baczewski, Andrew David, Vikram, Melapudi, Shanker, Balasubramaniam, & Kempel, Leo. Accelerated Cartesian expansion (ACE) based framework for the rapid evaluation of diffusion, lossy wave, and Klein-Gordon potentials. United States. doi:10.1016/j.jcp.2010.08.025.
```

```
Baczewski, Andrew David, Vikram, Melapudi, Shanker, Balasubramaniam, and Kempel, Leo. Fri .
"Accelerated Cartesian expansion (ACE) based framework for the rapid evaluation of diffusion, lossy wave, and Klein-Gordon potentials". United States. doi:10.1016/j.jcp.2010.08.025. https://www.osti.gov/servlets/purl/1252698.
```

```
@article{osti_1252698,
```

title = {Accelerated Cartesian expansion (ACE) based framework for the rapid evaluation of diffusion, lossy wave, and Klein-Gordon potentials},

author = {Baczewski, Andrew David and Vikram, Melapudi and Shanker, Balasubramaniam and Kempel, Leo},

abstractNote = {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(Ns2Nt2), where Ns and Nt 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(NsNtlog2Nt). Furthermore, several numerical results are presented for the diffusion equation to validate the accuracy and efficacy of the fast algorithms developed here.},

doi = {10.1016/j.jcp.2010.08.025},

journal = {Journal of Computational Physics},

number = 24,

volume = 229,

place = {United States},

year = {2010},

month = {8}

}

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