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Title: 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 » section discusses the complexity of both methods in comparison with naive evaluation of a retarded-time integral.« less

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.
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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
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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.
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@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}
}
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Journal Article:
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Publisher's Version of Record
https://doi.org/10.1016/j.cam.2017.04.007
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Figures / Tables:

Figure 1 Figure 1: Retarded-time integration in terms of bunch history. In the right-hand figure the $z$-coordinate normal to the sheet is suppressed.
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Works referenced in this record:

Stability and Convergence of Collocation Schemes for Retarded Potential Integral Equations
journal, January 2004

Nonreflecting Boundary Conditions for the Time-Dependent Wave Equation
journal, July 2002

  • Alpert, Bradley; Greengard, Leslie; Hagstrom, Thomas
  • Journal of Computational Physics, Vol. 180, Issue 1
  • DOI: 10.1006/jcph.2002.7093

Rapid Evaluation of Nonreflecting Boundary Kernels for Time-Domain Wave Propagation
journal, January 2000

  • Alpert, Bradley; Greengard, Leslie; Hagstrom, Thomas
  • SIAM Journal on Numerical Analysis, Vol. 37, Issue 4
  • DOI: 10.1137/S0036142998336916

A Bootstrap Method for Sum-of-Poles Approximations
journal, June 2012

Microbunching instability in a chicane: Two-dimensional mean field treatment
journal, August 2009

  • Bassi, Gabriele; Ellison, James A.; Heinemann, Klaus
  • Physical Review Special Topics - Accelerators and Beams, Vol. 12, Issue 8
  • DOI: 10.1103/PhysRevSTAB.12.080704

Radiation boundary conditions for time-dependent waves based on complete plane wave expansions
journal, July 2010

  • Hagstrom, Thomas; Warburton, Timothy; Givoli, Dan
  • Journal of Computational and Applied Mathematics, Vol. 234, Issue 6
  • DOI: 10.1016/j.cam.2009.08.050

On approximation of functions by exponential sums
journal, July 2005

Complete Radiation Boundary Conditions: Minimizing the Long Time Error Growth of Local Methods
journal, January 2009

  • Hagstrom, Thomas; Warburton, Timothy
  • SIAM Journal on Numerical Analysis, Vol. 47, Issue 5
  • DOI: 10.1137/090745477

On partial spherical means formulas and radiation boundary conditions for the 3+1 wave equation
journal, February 2010

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    Figure 1 (p. 4)figure
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