Rapid evaluation of twodimensional retarded time integrals
Abstract
We present two methods for rapid evaluation of twodimensional 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 twodimensional 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 Laplacetransform 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 methods are relevant for the sheetbunch formulation of the Vlasov–Maxwell system,more »
 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) (SC25)
 OSTI Identifier:
 1511845
 Alternate Identifier(s):
 OSTI ID: 1416621; OSTI ID: 1530304
 Grant/Contract Number:
 FG0299ER41104; FG0399ER41104; FG99ER41104; AC0205CH11231
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of Computational and Applied Mathematics
 Additional Journal Information:
 Journal Volume: 324; Journal Issue: C; Journal ID: ISSN 03770427
 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 twodimensional 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 twodimensional retarded time integrals. United States. doi: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 twodimensional retarded time integrals". United States. doi:https://doi.org/10.1016/j.cam.2017.04.007. https://www.osti.gov/servlets/purl/1511845.
@article{osti_1511845,
title = {Rapid evaluation of twodimensional 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 twodimensional 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 twodimensional 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 Laplacetransform 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 sheetbunch 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 retardedtime 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}
}
Web of Science
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