# Fast Evaluation of Far-Field Signals for Time-Domain Wave Propagation

- 103 Downloads
- 1 Citations

## Abstract

Time-domain simulation of wave phenomena on a finite computational domain often requires a fictitious outer boundary. An important practical issue is the specification of appropriate boundary conditions on this boundary, often conditions of complete transparency. Attention to this issue has been paid elsewhere, and here we consider a different, although related, issue: far-field signal recovery. Namely, from smooth data recorded on the outer boundary we wish to recover the far-field signal which would reach arbitrarily large distances. These signals encode information about interior scatterers and often correspond to actual measurements. This article expresses far-field signal recovery in terms of time-domain convolutions, each between a solution multipole moment recorded at the boundary and a sum-of-exponentials kernel. Each exponential corresponds to a pole term in the Laplace transform of the kernel, a finite sum of simple poles. Greengard, Hagstrom, and Jiang have derived the large-\(\ell \) (spherical-harmonic index) asymptotic expansion for the pole residues, and their analysis shows that, when expressed in terms of the exact sum-of-exponentials, large-\(\ell \) signal recovery is plagued by cancellation errors. Nevertheless, through an alternative integral representation of the kernel and its subsequent approximation by a *smaller* number of exponential terms (kernel compression), we are able to alleviate these errors and achieve accurate signal recovery. We empirically examine scaling relations between the parameters which determine a compressed kernel, and perform numerical tests of signal “teleportation” from one radial value \(r_1\) to another \(r_2\), including the case \(r_2=\infty \). We conclude with a brief discussion on application to other hyperbolic equations posed on non-flat geometries where waves undergo backscatter.

## Keywords

Pole Location Radiation Boundary Condition Quadruple Precision Cancellation Error Precision Format## Notes

### Acknowledgments

SRL gratefully acknowledges support from NSF grant No. PHY 0855678 to the University of New Mexico, with which infrastructure for our approximations was developed. SEF acknowledges support from the Joint Space Science Institute and NSF Grants No. PHY 1208861 and No. PHY 1005632 to the University of Maryland, NSF Grants PHY-1306125 and AST-1333129 to Cornell University, and by a grant from the Sherman Fairchild Foundation. For insights and helpful comments we wish to thank Thomas Hagstrom and Akil Narayan.

## References

- 1.Wilcox, C.H.: The initial-boundary value problem for the wave equation in an exterior domain with spherical boundary. Notices Am. Math. Soc.
**6**, 869–870 (1959)Google Scholar - 2.Tokita, T.: Exponential decay of solutions for the wave equation in the exterior domain with spherical boundary. J. Math. Kyoto Univ.
**12–2**, 413–430 (1972)MathSciNetGoogle Scholar - 3.Benedict, A.G., Field, S.E., Lau, S.R.: Fast evaluation of asymptotic waveforms from gravitational perturbations. Class. Quantum Grav.
**30**, 055015 (2013)MathSciNetCrossRefGoogle Scholar - 4.Greengard, L., Hagstrom, T., Jiang, S.: The solution of the scalar wave equation in the exterior of a sphere, J. Comput. Phys.
**274**, 191–207 (2014). arXiv:1308.0643 - 5.Grote, M.J., Keller, J.B.: Nonreflecting boundary conditions for time-dependent scattering. J. Comput. Phys.
**127**(1), 52–65 (1996)MathSciNetCrossRefMATHGoogle Scholar - 6.Alpert, B., Greengard, L., Hagstrom, T.: Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation. SIAM J. Numer. Anal.
**37**, 1138–1164 (2000)MathSciNetCrossRefMATHGoogle Scholar - 7.Jiang, S.: Fast Evaluation of Nonreflecting Boundary Conditions for the Schrödinger Equation, New York University, Ph. D. Dissertation (2001)Google Scholar
- 8.Xu, K., Jiang, S.: A bootstrap method for sum-of-poles approximations. J. Sci. Comput.
**55**(1), 16–39 (2013)MathSciNetCrossRefMATHGoogle Scholar - 9.Bizzozero, D. A., Ellison, J. A., Heinemann, K., Lau, S.R.: Rapid evaluation of two-dimensional retarded time integrals. (2015, in preparation)Google Scholar
- 10.Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974)Google Scholar
- 11.Olver, F.W.J.: The asymptotic expansion of Bessel functions of large order. Philos. Trans. R. Soc. Lond. A
**247**, 328–368 (1954)MathSciNetCrossRefGoogle Scholar - 12.Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover Publishing Inc., New York (1970)Google Scholar
- 13.Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1944)MATHGoogle Scholar
- 14.Thompson, I.J., Barnett, A.R.: Coulomb and Bessel functions of complex arguments and order. J. Comp. Phys.
**64**(2), 490–509 (1986)MathSciNetCrossRefMATHGoogle Scholar - 15.Field, S.E., Hesthaven, J.S., Lau, S.R.: Persistent junk solutions in time-domain modeling of extreme mass ratio binaries. Phys. Rev.
**D81**, 124030 (2010)Google Scholar - 16.Lau, S.R.: Analytic structure of radiation boundary kernels for blackhole perturbations. J. Math. Phys.
**46**, 102503 (2005)MathSciNetCrossRefGoogle Scholar - 17.Lau, S.R.: Rapid evaluation of radiation boundary kernels for time-domain wave propagation on blackholes: theory and numerical methods. J. Comput. Phys.
**199**(1), 376–422 (2004)MathSciNetCrossRefMATHGoogle Scholar - 18.Kernels are available at either of the following websites. www.math.unm.edu/~lau/KernelsRWZ; www.dam.brown.edu/people/sfield/KernelsRWZ