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A Fixed-Point Iteration Method for High Frequency Helmholtz Equations

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Abstract

For numerically solving the high frequency Helmholtz equation, the conventional finite difference and finite element methods based on discretizing the equation on meshes usually suffer from the numerical dispersion errors (‘pollution effect’) that require very refined meshes (Babuska and Sauter in SIAM Rev 42(3):451–484, 2000). Asymptotic methods like geometrical optics provide an alternative way to compute the solutions without ‘pollution effect’, but they generally can only compute locally valid approximations for the solutions and fail to capture the caustics faithfully. In order to obtain globally valid solutions efficiently without ‘pollution effect’, we transfer the problem into a fixed-point problem related to an exponential operator, and the associated functional evaluations are achieved by unconditionally stable operator-splitting based pseudospectral schemes such that large step sizes are allowed to reach the approximated fixed point efficiently for certain prescribed accuracy requirement. And the Anderson acceleration is incorporated to accelerate the convergence. Both two-dimensional and three-dimensional numerical experiments are presented to demonstrate the method.

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References

  1. Aziz, A.K., Kellogg, R.B., Stephens, A.B.: A two point boundary value problem with a rapidly oscillating solution. Numerische Mathematik 53, 107–121 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Babich, V.M.: The short wave asymptotic form of the solution for the problem of a point source in an inhomogeneous medium. USSR Comput. Math. Math. Phys. 5(5), 247–251 (1965)

    Article  MATH  Google Scholar 

  3. Babuska, I.M., Sauter, S.A.: Is the pollution effect of the fem avoidable for the Helmholtz equation considering high wave numbers? SIAM Rev. 42(3), 451–484 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Bao, W., Jaksch, D., Markowich, P.A.: Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation. J. Comput. Phys. 187(1), 318–342 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berenger, J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114(2), 185–200 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bourgeois, A., Bourget, M., Lailly, P., Poulet, M., Ricarte, P., Versteeg, R.: Marmousi, model and data. In: The Marmousi Experience, pp. 5–16 (1991)

  7. Castella, F., Chartier, P., Descombes, S., Vilmart, G.: Splitting methods with complex times for parabolic equations. BIT Numer. Math. 49(3), 487–508 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Červený, V., Popov, M.M., Pšenčík, I.: Computation of wave fields in inhomogeneous media: Gaussian beam approach. Geophys. J. R. Astron. Soc. 70(1), 109–128 (1982)

    Article  MATH  Google Scholar 

  9. Engquist, B., Runborg, O.: Computational high frequency wave propagation. Acta Numerica 12, 181–266 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Erlangga, Y.A., Vuik, C., Oosterlee, C.W.: Comparison of multigrid and incomplete Lu shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation. Appl. Numer. Math. 56(5), 648–666 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Geoltrain, S., Brac, J.: Can we image complex structures with first-arrival traveltime. Geophysics 58, 564–575 (1993)

    Article  Google Scholar 

  12. Gray, S., May, W.: Kirchhoff migration using Eikonal equation traveltimes. Geophysics 59, 810–817 (1994)

    Article  Google Scholar 

  13. Jacobs, M., Luo, S.: Asymptotic solutions for high frequency Helmholtz equations in anisotropic media with Hankel functions. J. Sci. Comput. 80(2), 808–833 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jacobs, M., Luo, S.: Numerical solutions for point-source high frequency Helmholtz equation through efficient time propagators for Schrödinger equation. J. Comput. Phys. 438, 110357 (2021)

    Article  MATH  Google Scholar 

  15. Keller, J.B., Lewis, R.M.: Asymptotic methods for partial differential equations: the reduced wave equation and Maxwell’s equations. Surv. Appl. Math. 1, 1–82 (1995)

    MathSciNet  MATH  Google Scholar 

  16. Liu, Q.H.: The PSTD algorithm: a time-domain method requiring only two cells per wavelength. Microw. Opt. Technol. Lett. 15(3), 158–165 (1997)

    Article  Google Scholar 

  17. Liu, Q.H.: Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm. IEEE Trans. Geosci. Remote Sens. 37(2), 917–926 (1999)

    Article  Google Scholar 

  18. Lu, W., Qian, J., Burridge, R.: Babich’s expansion and the fast Huygens sweeping method for the Helmholtz wave equation at high frequencies. J. Comput. Phys. 313, 478–510 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ludwig, D.: Uniform asymptotic expansions at a caustic. Commun. Pure Appl. Math. 19(2), 215–250 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  20. Luo, S., Qian, J.: Factored singularities and high-order Lax–Friedrichs sweeping schemes for point-source traveltimes and amplitudes. J. Comput. Phys. 230(12), 4742–4755 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Luo, S., Qian, J., Burridge, R.: Fast Huygens sweeping methods for Helmholtz equations in inhomogeneous media in the high frequency regime. J. Comput. Phys. 270, 378–401 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Luo, S., Qian, J., Zhao, H.: Higher-order schemes for 3D first-arrival traveltimes and amplitudes. Geophysics 77(2), T47–T56 (2012)

    Article  Google Scholar 

  23. Manolopoulos, D.E.: Derivation and reflection properties of a transmission-free absorbing potential. J. Chem. Phys. 117(21), 9552–9559 (2002)

    Article  Google Scholar 

  24. Maslov, V.P., Fedoriuk, M.V.: Semi-classical Approximation in Quantum Mechanics. D. Reidel Publishing Company, Dordrecht (1981)

    Book  Google Scholar 

  25. Muga, J.G., Palao, J.P., Navarro, B., Egusquiza, I.L.: Complex absorbing potentials. Phys. Rep. 395(6), 357–426 (2004)

    Article  MathSciNet  Google Scholar 

  26. Nichols, D.: Imaging complex structures using band limited Green’s functions. Ph.D. thesis, Stanford University, Stanford, CA94305 (1994)

  27. Qian, J., Lu, W., Yuan, L., Luo, S., Burridge, R.: Eulerian geometrical optics and fast Huygens sweeping methods for three-dimensional time-harmonic high-frequency Maxwell’s equations in inhomogeneous media. Multiscale Model. Simul. 14(2), 595–636 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Qian, J., Luo, S., Burridge, R.: Fast Huygens’ sweeping methods for multiarrival Green’s functions of Helmholtz equations in the high-frequency regime. Geophysics 80(2), T91–T100 (2015)

    Article  Google Scholar 

  29. Qian, J., Yuan, L., Liu, Y., Luo, S., Burridge, R.: Babich’s expansion and high-order Eulerian asymptotics for point-source Helmholtz equations. J. Sci. Comput. 67(3), 883–908 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. SEG/EAGE, SEG/EAGE Salt and Overthrust Models, https://wiki.seg.org/wiki/SEG/EAGE_Salt_and_Overthrust_Models, (1997)

  31. Söderlind, G.: Automatic control and adaptive time-stepping. Numer. Algorithms 31(1), 281–310 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  32. Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5(3), 506–517 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  33. Walker, H.F., Ni, P.: Anderson acceleration for fixed-point iterations. SIAM J. Numer. Anal. 49(4), 1715–1735 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wood, W.L.: Practical Time-Stepping Schemes. Oxford Applied Mathematics and Computing Science Series. Clarendon Press, Oxford University Press, Oxford (1990)

    Google Scholar 

  35. Yu, Y., Esry, B.D.: An optimized absorbing potential for ultrafast, strong-field problems. arXiv Computational Physics (2017)

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Funding

This work was partially supported by NSF DMS 1719907.

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Authors and Affiliations

  1. Department of Mathematics, Iowa State University, Ames, IA, 50011, USA

    Songting Luo

  2. Department of Electrical and Computer Engineering, Duke University, Box 90291, Durham, NC, 27708-0291, USA

    Qing Huo Liu

Authors
  1. Songting Luo
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  2. Qing Huo Liu
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Contributions

QHL: Conceptualization, Methodology, Writing—Review and Editing; SL: Conceptualization, Methodology, Formal analysis, Visualization, Validation, Writing—Original Draft , Writing—Review and Editing.

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Correspondence to Songting Luo.

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This work was partially supported by National Science Foundation Division of Mathematical Sciences 1719907.

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Luo, S., Liu, Q.H. A Fixed-Point Iteration Method for High Frequency Helmholtz Equations. J Sci Comput 93, 74 (2022). https://doi.org/10.1007/s10915-022-02039-8

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  • DOI: https://doi.org/10.1007/s10915-022-02039-8

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