# Rapid evaluation of two-dimensional retarded time integrals

## Abstract

We present two methods for rapid evaluation of two-dimensional retarded time integrals. For example, such integrals arise as the z = 0 trace *U*( *t*, *x*, *y*, 0) of a solution *U*( *t*, *x*, *y*, *z*) to 3 + 1 wave equation *U* = –2 *f*( *t*, *x*, *y*)δ( *z*) forced by a ‘‘sheet source’’ at *z* = 0. The spatial Fourier transform of a two-dimensional retarded time integral involves a temporal convolution with the zeroth order Bessel function *J* _{0}( *t*). Appealing to work by Alpert, Greengard, and Hagstrom and by Xu and Jiang on rational approximation in the Laplace-transform domain, our first method relies on approximation of *J* _{0}( *t*) as a sum of exponential functions. We achieve approximations with double precision accuracy near *t* ≃ 0, and maintain single precision accuracy out to *T* ≃ 10 ^{8}. Our second method involves evolution of the 3 + 1 wave equation in a ‘‘thin block’’ above the sheet, adopting the radiation boundary conditions of Hagstrom, Warburton, and Givoli based on complete plane wave expansions. We review their technique, present its implementation for our problem, and present new results on the nonlocal spacetime form of radiation boundary conditions. Our methodsmore »

- Authors:

- Univ. of New Mexico, Albuquerque, NM (United States)

- Publication Date:

- Research Org.:
- Univ. of New Mexico, Albuquerque, NM (United States); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1511845

- Alternate Identifier(s):
- OSTI ID: 1416621; OSTI ID: 1530304

- Grant/Contract Number:
- FG02-99ER41104; FG03-99ER41104; FG-99ER41104; AC02-05CH11231

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 324; Journal Issue: C; Journal ID: ISSN 0377-0427

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 98 NUCLEAR DISARMAMENT, SAFEGUARDS, AND PHYSICAL PROTECTION; Retarded time integral; Rational approximation; Radiation boundary conditions; Initial boundary value problem; Vlasov–Maxwell system; Accelerator beam physics

### Citation Formats

```
Bizzozero, D. A., Ellison, J. A., Heinemann, K., and Lau, S. R. Rapid evaluation of two-dimensional retarded time integrals. United States: N. p., 2017.
Web. doi:10.1016/j.cam.2017.04.007.
```

```
Bizzozero, D. A., Ellison, J. A., Heinemann, K., & Lau, S. R. Rapid evaluation of two-dimensional retarded time integrals. United States. doi:10.1016/j.cam.2017.04.007.
```

```
Bizzozero, D. A., Ellison, J. A., Heinemann, K., and Lau, S. R. Wed .
"Rapid evaluation of two-dimensional retarded time integrals". United States. doi:10.1016/j.cam.2017.04.007. https://www.osti.gov/servlets/purl/1511845.
```

```
@article{osti_1511845,
```

title = {Rapid evaluation of two-dimensional retarded time integrals},

author = {Bizzozero, D. A. and Ellison, J. A. and Heinemann, K. and Lau, S. R.},

abstractNote = {We present two methods for rapid evaluation of two-dimensional retarded time integrals. For example, such integrals arise as the z = 0 trace U(t, x, y, 0) of a solution U(t, x, y, z) to 3 + 1 wave equation U = –2f(t, x, y)δ(z) forced by a ‘‘sheet source’’ at z = 0. The spatial Fourier transform of a two-dimensional retarded time integral involves a temporal convolution with the zeroth order Bessel function J0(t). Appealing to work by Alpert, Greengard, and Hagstrom and by Xu and Jiang on rational approximation in the Laplace-transform domain, our first method relies on approximation of J0(t) as a sum of exponential functions. We achieve approximations with double precision accuracy near t ≃ 0, and maintain single precision accuracy out to T ≃ 108. Our second method involves evolution of the 3 + 1 wave equation in a ‘‘thin block’’ above the sheet, adopting the radiation boundary conditions of Hagstrom, Warburton, and Givoli based on complete plane wave expansions. We review their technique, present its implementation for our problem, and present new results on the nonlocal spacetime form of radiation boundary conditions. Our methods are relevant for the sheet-bunch formulation of the Vlasov–Maxwell system, although here we only test methods on a model problem, a Gaussian source following an elliptical orbit. Here, our concluding section discusses the complexity of both methods in comparison with naive evaluation of a retarded-time integral.},

doi = {10.1016/j.cam.2017.04.007},

journal = {Journal of Computational and Applied Mathematics},

number = C,

volume = 324,

place = {United States},

year = {2017},

month = {4}

}

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