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=-2ƒ(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. Our concludingmore »
- Authors:
- 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)
- OSTI Identifier:
- 1828433
- Alternate Identifier(s):
- OSTI ID: 1416621; OSTI ID: 1511845; OSTI ID: 1530304
- Grant/Contract Number:
- FG-99ER41104; FG02-99ER41104; FG03-99ER41104; AC02-05CH11231
- Resource Type:
- Published Article
- Journal Name:
- Journal of Computational and Applied Mathematics
- Additional Journal Information:
- Journal Name: Journal of Computational and Applied Mathematics Journal Volume: 324 Journal Issue: C; Journal ID: ISSN 0377-0427
- Publisher:
- Elsevier
- Country of Publication:
- Belgium
- 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. Belgium: 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. Belgium. https://doi.org/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". Belgium. https://doi.org/10.1016/j.cam.2017.04.007.
@article{osti_1828433,
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=-2ƒ(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. 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 = {Belgium},
year = {Wed Nov 01 00:00:00 EDT 2017},
month = {Wed Nov 01 00:00:00 EDT 2017}
}
https://doi.org/10.1016/j.cam.2017.04.007
Web of Science
Figures / Tables:
Works referenced in this record:
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Figures / Tables found in this record: