Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 
 
 

 

Error estimates for Gaussian beam superpositions


Authors: Hailiang Liu, Olof Runborg and Nicolay M. Tanushev
Journal: Math. Comp. 82 (2013), 919-952
MSC (2010): Primary 35J10, 35L05, 35A35, 41A60, 35L30
DOI: https://doi.org/10.1090/S0025-5718-2012-02656-1
Published electronically: November 6, 2012
MathSciNet review: 3008843
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Gaussian beams are asymptotically valid high frequency solutions to hyperbolic partial differential equations, concentrated on a single curve through the physical domain. They can also be extended to some dispersive wave equations, such as the Schrödinger equation. Superpositions of Gaussian beams provide a powerful tool to generate more general high frequency solutions that are not necessarily concentrated on a single curve. This work is concerned with the accuracy of Gaussian beam superpositions in terms of the wavelength $ \varepsilon $. We present a systematic construction of Gaussian beam superpositions for all strictly hyperbolic and Schrödinger equations subject to highly oscillatory initial data of the form $ Ae^{i\Phi /\varepsilon }$. Through a careful estimate of an oscillatory integral operator, we prove that the $ k$-th order Gaussian beam superposition converges to the original wave field at a rate proportional to $ \varepsilon ^{k/2}$ in the appropriate norm dictated by the well-posedness estimate. In particular, we prove that the Gaussian beam superposition converges at this rate for the acoustic wave equation in the standard, $ \varepsilon $-scaled, energy norm and for the Schrödinger equation in the $ L^2$ norm. The obtained results are valid for any number of spatial dimensions and are unaffected by the presence of caustics. We present a numerical study of convergence for the constant coefficient acoustic wave equation in $ \mathbb{R}^2$ to analyze the sharpness of the theoretical results.


References [Enhancements On Off] (What's this?)

  • 1. G. Ariel, B. Engquist, N. M. Tanushev, and R. Tsai.
    Gaussian beam decomposition of high frequency wave fields using expectation-maximization.
    J. Comput. Phys., 230(6):2303-2321, 2011. MR 2764548 (2011j:78003)
  • 2. V. M. Babič and V. S. Buldyrev.
    Short-Wavelength Diffraction Theory: Asymptotic Methods, volume 4 of Springer Series on Wave Phenomena.
    Springer-Verlag, 1991. MR 1245488 (94f:78004)
  • 3. V. M. Babič and T. F. Pankratova.
    On discontinuities of Green's function of the wave equation with variable coefficient.
    Problemy Matem. Fiziki, 6, 1973.
    Leningrad University, Saint-Petersburg. MR 0355352 (50:7826)
  • 4. V. M. Babič and M. M. Popov.
    Gaussian summation method (review).
    Izv. Vyssh. Uchebn. Zaved. Radiofiz., 32(12):1447-1466, 1989. MR 1045524 (91c:78004)
  • 5. S. Bougacha, J.L. Akian, and R. Alexandre.
    Gaussian beams summation for the wave equation in a convex domain.
    Commun. Math. Sci., 7(4):973-1008, 2009. MR 2604628 (2011b:35284)
  • 6. V. Červený, M. M. Popov, and I. Pšenčík.
    Computation of wave fields in inhomogeneous media -- Gaussian beam approach.
    Geophys. J. R. Astr. Soc., 70:109-128, 1982.
  • 7. B. Engquist and O. Runborg.
    Computational high frequency wave propagation.
    Acta Numerica, 12:181-266, 2003. MR 2249156 (2007f:65043)
  • 8. N. R. Hill.
    Gaussian beam migration.
    Geophysics, 55(11):1416-1428, 1990.
  • 9. N. R. Hill.
    Prestack Gaussian beam depth migration.
    Geophysics, 66(4):1240-1250, 2001.
  • 10. L. Hörmander.
    Fourier integral operators. I.
    Acta Math., 127(1-2):79-183, 1971. MR 0388463 (52:9299)
  • 11. L. Hörmander.
    On the existence and the regularity of solutions of linear pseudo-differential equations.
    L'Enseignement Mathématique, XVII:99-163, 1971. MR 0331124 (48:9458)
  • 12. L. Hörmander.
    The analysis of linear partial differential operators. I.
    Classics in Mathematics. Springer, Berlin, 2003.
    Distribution theory and Fourier analysis, Reprint of the 1990 edition. MR 1996773
  • 13. L. Hörmander.
    The analysis of linear partial differential operators. III.
    Classics in Mathematics. Springer, Berlin, 2007.
    Pseudo-differential operators, Reprint of the 1994 edition. MR 2304165 (2007k:35006)
  • 14. S. Jin, H. Wu, and X. Yang.
    Gaussian beam methods for the Schrödinger equation in the semi-classical regime: Lagrangian and Eulerian formulations.
    Commun. Math. Sci., 6:995-1020, 2008. MR 2511703 (2010f:65217)
  • 15. A. P. Katchalov and M. M. Popov.
    Application of the method of summation of Gaussian beams for calculation of high-frequency wave fields.
    Sov. Phys. Dokl., 26:604-606, 1981.
  • 16. J. B. Keller.
    Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems.
    Ann. Physics, 4:180-188, 1958. MR 0099207 (20:5650)
  • 17. L. Klimeš.
    Expansion of a high-frequency time-harmonic wavefield given on an initial surface into Gaussian beams.
    Geophys. J. R. Astr. Soc., 79:105-118, 1984.
  • 18. L. Klimeš.
    Discretization error for the superposition of Gaussian beams.
    Geophys. J. R. Astr. Soc., 86:531-551, 1986.
  • 19. Yu. A. Kravtsov.
    On a modification of the geometrical optics method.
    Izv. VUZ Radiofiz., 7(4):664-673, 1964.
  • 20. S. Leung and J. Qian.
    Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime.
    J. Comput. Phys., 228:2951-2977, 2009. MR 2509304 (2010e:81093)
  • 21. S. Leung, J. Qian, and R. Burridge.
    Eulerian Gaussian beams for high frequency wave propagation.
    Geophysics, 72:SM61-SM76, 2007.
  • 22. H. Liu and J. Ralston.
    Recovery of high frequency wave fields for the acoustic wave equation.
    Multiscale Model. Sim., 8(2):428-444, 2009. MR 2581028 (2011a:35298)
  • 23. H. Liu and J. Ralston.
    Recovery of high frequency wave fields from phase space-based measurements.
    Multiscale Model. Sim., 8(2):622-644, 2010. MR 2600299 (2011a:35438)
  • 24. D. Ludwig.
    Uniform asymptotic expansions at a caustic.
    Comm. Pure Appl. Math., 19:215-250, 1966. MR 0196254 (33:4446)
  • 25. V. P. Maslov and M. V. Fedoriuk.
    Semiclassical approximation in quantum mechanics, volume 7 of Mathematical Physics and Applied Mathematics.
    D. Reidel Publishing Co., Dordrecht, 1981.
    Translated from the Russian by J. Niederle and J. Tolar, Contemporary Mathematics, 5. MR 634377 (84k:58226)
  • 26. M. Motamed and O. Runborg.
    A wave front Gaussian beam method for high-frequency wave propagation.
    In Proceedings of WAVES 2007, University of Reading, UK, 2007.
  • 27. M. Motamed and O. Runborg.
    Taylor expansion and discretization errors in Gaussian beam superposition.
    Wave Motion, 2010. MR 2684020 (2011g:65292)
  • 28. M. M. Popov.
    A new method of computation of wave fields using Gaussian beams.
    Wave Motion, 4:85-97, 1982. MR 638867 (82m:73021)
  • 29. J. Qian and L. Ying.
    Fast Gaussian wavepacket transforms and Gaussian beams for the Schrödinger equation.
    Journal of Computational Physics, 229:7848-7873, 2010. MR 2674307 (2011f:65226)
  • 30. J. Ralston.
    Gaussian beams and the propagation of singularities.
    In Studies in partial differential equations, volume 23 of MAA Stud. Math., pages 206-248. Math. Assoc. America, Washington, DC, 1982. MR 716507 (85c:35052)
  • 31. V. Rousse and T. Swart.
    Global $ {L}^2$-boundedness theorems for semiclassical Fourier integral operators with complex phase.
    arxiv:0710.4200v3, 2007.
  • 32. V. Rousse and T. Swart.
    A mathematical justification for the Herman-Kluk propagator.
    Comm. Math. Phys., 286(2):725-750, 2009. MR 2472042 (2010a:81066)
  • 33. O. Runborg.
    Mathematical models and numerical methods for high frequency waves.
    Commun. Comput. Phys., 2:827-880, 2007. MR 2355631 (2008j:78007)
  • 34. N. M. Tanushev.
    Superpositions and higher order Gaussian beams.
    Commun. Math. Sci., 6(2):449-475, 2008. MR 2433704 (2010a:35008)
  • 35. N. M. Tanushev, B. Engquist, and R. Tsai.
    Gaussian beam decomposition of high frequency wave fields.
    J. Comput. Phys., 228(23):8856-8871, 2009. MR 2558781 (2010k:78001)
  • 36. N. M. Tanushev, J. Qian, and J. V. Ralston.
    Mountain waves and Gaussian beams.
    Multiscale Model. Simul., 6(2):688-709, 2007. MR 2338499 (2008i:76046)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 35J10, 35L05, 35A35, 41A60, 35L30

Retrieve articles in all journals with MSC (2010): 35J10, 35L05, 35A35, 41A60, 35L30


Additional Information

Hailiang Liu
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Email: hliu@iastate.edu

Olof Runborg
Affiliation: Department of Numerical Analysis, CSC, KTH, 100 44 Stockholm, Sweden – and – Swedish e-Science Research Center (SeRC), KTH, 100 44 Stockholm, Sweden
Email: olofr@nada.kth.se

Nicolay M. Tanushev
Affiliation: Department of Mathematics, The University of Texas at Austin, 1 University Station, C1200, Austin, Texas 78712
Address at time of publication: Z-Terra Inc., 17171 Park Row, Suite 247, Houston, Texas 77084
Email: nicktan@math.utexas.edu

DOI: https://doi.org/10.1090/S0025-5718-2012-02656-1
Keywords: High-frequency wave propagation, error estimates, Gaussian beams
Received by editor(s): August 6, 2010
Received by editor(s) in revised form: June 2, 2011, and August 31, 2011
Published electronically: November 6, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society