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An optimal 13-point finite difference scheme for a 2D Helmholtz equation with a perfectly matched layer boundary condition

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Abstract

Efficient and accurate numerical schemes for solving the Helmholtz equation are critical to the success of various wave propagation–related inverse problems, for instance, the full-waveform inversion problem. However, the numerical solution to a multi-dimensional Helmholtz equation is notoriously difficult, especially when a perfectly matched layer (PML) boundary condition is incorporated. In this paper, an optimal 13-point finite difference scheme for the Helmholtz equation with a PML in the two-dimensional domain is presented. An error analysis for the numerical approximation of the exact wavenumber is provided. Based on error analysis, the optimal 13-point finite difference scheme is developed so that the numerical dispersion is minimized. Two practical strategies for selecting optimal parameters are presented. Several numerical examples are solved by the new method to illustrate its accuracy and effectiveness in reducing numerical dispersion.

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References

  1. Alford, R.M., Kelly, K.R., Boore, D.: Accuracy of finite-difference modeling of the acoustic wave equation. Geophysics 39(6), 834–842 (1974)

    Article  Google Scholar 

  2. Bayliss, A., Goldstein, C.I., Turkel, E.: On accuracy conditions for the numerical computation of waves. J. Comput. Phys. 59(3), 396–404 (1985)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Britt, S., Tsynkov, S., Turkel, E.: Numerical simulation of time-harmonic waves in inhomogeneous media using compact high order schemes. Commun. Comput. Phys. 9(3), 520–541 (2011)

    Article  MathSciNet  Google Scholar 

  5. Chen, J.B.: A generalized optimal 9-point scheme for frequency-domain scalar wave equation. J. Appl. Geophys. 92, 1–7 (2013)

    Article  Google Scholar 

  6. Chen, Z., Cheng, D., Feng, W., Wu, T.: An optimal 9-point finite difference scheme for the Helmholtz equation with PML. International Journal of Numerical Analysis & Modeling 10(2) (2013)

  7. Chen, Z., Cheng, D., Wu, T.: A dispersion minimizing finite difference scheme and preconditioned solver for the 3D Helmholtz equation. J. Comput. Phys. 231(24), 8152–8175 (2012)

    Article  MathSciNet  Google Scholar 

  8. Cheng, D., Tan, X., Zeng, T.: A dispersion minimizing finite difference scheme for the Helmholtz equation based on point-weighting. Comput. Math. Appl. 73(11), 2345–2359 (2017)

    Article  MathSciNet  Google Scholar 

  9. Collino, F., Monk, P.B.: Optimizing the perfectly matched layer. Comput. Methods Appl. Mech. Eng. 164(1–2), 157–171 (1998)

    Article  MathSciNet  Google Scholar 

  10. Dastour, H., Liao, W.: A fourth-order optimal finite difference scheme for the Helmholtz equation with PML. Comput. Math. Appl. 6(78), 2147–2165 (2019)

    Article  MathSciNet  Google Scholar 

  11. Davis, T.A.: Algorithm 832: UMFPACK v4. 3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. (TOMS) 30(2), 196–199 (2004)

    Article  MathSciNet  Google Scholar 

  12. Davis, T.A., Duff, I.S.: An unsymmetric-pattern multifrontal method for sparse lu factorization. SIAM J. Matrix Anal. Appl. 18(1), 140–158 (1997)

    Article  MathSciNet  Google Scholar 

  13. De Zeeuw, P.M.: Matrix-dependent prolongations and restrictions in a blackbox multigrid solver. J. Comput. Appl. Math. 33(1), 1–27 (1990)

    Article  MathSciNet  Google Scholar 

  14. Engquist, B., Majda, A.: Absorbing boundary conditions for numerical simulation of waves. Proc. Natl. Acad. Sci. 74(5), 1765–1766 (1977)

    Article  MathSciNet  Google Scholar 

  15. Engquist, B., Ying, L.: Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers. Multiscale Model. Simul. 9(2), 686–710 (2011)

    Article  MathSciNet  Google Scholar 

  16. Erlangga, Y.A., Oosterlee, C.W., Vuik, C.: A novel multigrid based preconditioner for heterogeneous Helmholtz problems. SIAM J. Sci. Comput. 27(4), 1471–1492 (2006)

    Article  MathSciNet  Google Scholar 

  17. van Gijzen, M.B., Erlangga, Y.A., Vuik, C.: Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian. SIAM J. Sci. Comput. 29(5), 1942–1958 (2007)

    Article  MathSciNet  Google Scholar 

  18. Gordon, D., Gordon, R.: Carp-cg: A robust and efficient parallel solver for linear systems, applied to strongly convection dominated pdes. Parallel Comput. 36 (9), 495–515 (2010)

    Article  MathSciNet  Google Scholar 

  19. Harari, I., Turkel, E.: Accurate finite difference methods for time-harmonic wave propagation. J. Comput. Phys. 119(2), 252–270 (1995)

    Article  MathSciNet  Google Scholar 

  20. Ihlenburg, F., Babuška, I.: Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation. Int. J. Numer. Methods Eng. 38 (22), 3745–3774 (1995)

    Article  MathSciNet  Google Scholar 

  21. Ihlenburg, F., Babuška, I.: Finite element solution of the Helmholtz equation with high wave number part i: The h-version of the fem. Comput. Math. Appl. 30 (9), 9–37 (1995)

    Article  MathSciNet  Google Scholar 

  22. Jo, C.H., Shin, C., Suh, J.H.: An optimal 9-point, finite-difference, frequency-space, 2-D scalar wave extrapolator. Geophysics 61(2), 529–537 (1996)

    Article  Google Scholar 

  23. Medvinsky, M., Turkel, E., Hetmaniuk, U.: Local absorbing boundary conditions for elliptical shaped boundaries. J. Comput. Phys. 227(18), 8254–8267 (2008)

    Article  MathSciNet  Google Scholar 

  24. Nabavi, M., Siddiqui, M.K., Dargahi, J.: A new 9-point sixth-order accurate compact finite-difference method for the Helmholtz equation. J. Sound Vib. 307(3–5), 972–982 (2007)

    Article  Google Scholar 

  25. Operto, S., Virieux, J., Amestoy, P., L’Excellent, J.Y., Giraud, L., Ali, H.B.H.: 3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver: A feasibility study. Geophysics 72(5), SM195–SM211 (2007)

    Article  Google Scholar 

  26. Poulson, J., Engquist, B., Li, S., Ying, L.: A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations. SIAM J. Sci. Comput. 35 (3), C194–C212 (2013)

    Article  MathSciNet  Google Scholar 

  27. Pratt, R.G., Worthington, M.H.: Inverse theory applied to multi-source cross-hole tomography. Part 1: Acoustic wave-equation method. Geophys. Prospect. 38(3), 287–310 (1990)

    Article  Google Scholar 

  28. Shin, C., Sohn, H.: A frequency-space 2-D scalar wave extrapolator using extended 25-point finite-difference operator. Geophysics 63(1), 289–296 (1998)

    Article  Google Scholar 

  29. Singer, I., Turkel, E.: High-order finite difference methods for the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 163(1–4), 343–358 (1998)

    Article  MathSciNet  Google Scholar 

  30. Singer, I., Turkel, E.: A perfectly matched layer for the Helmholtz equation in a semi-infinite strip. J. Comput. Phys. 201(2), 439–465 (2004)

    Article  MathSciNet  Google Scholar 

  31. Sutmann, G.: Compact finite difference schemes of sixth order for the Helmholtz equation. J. Comput. Appl. Math. 203(1), 15–31 (2007)

    Article  MathSciNet  Google Scholar 

  32. Trefethen, L.N.: Group velocity in finite difference schemes. SIAM Rev. 24(2), 113–136 (1982)

    Article  MathSciNet  Google Scholar 

  33. Turkel, E., Gordon, D., Gordon, R., Tsynkov, S.: Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number. J. Comput. Phys. 232(1), 272–287 (2013)

    Article  MathSciNet  Google Scholar 

  34. Turkel, E., Yefet, A.: Absorbing PML boundary layers for wave-like equations. Appl. Numer. Math. 27(4), 533–557 (1998)

    Article  MathSciNet  Google Scholar 

  35. Virieux, J., Operto, S.: An overview of full-waveform inversion in exploration geophysics. Geophysics 74(6), WCC1–WCC26 (2009)

    Article  Google Scholar 

  36. Wang, S.: An improved high order finite difference method for non-conforming grid interfaces for the wave equation. J. Sci. Comput. 77(2), 775–792 (2018)

    Article  MathSciNet  Google Scholar 

  37. Wu, T.: A dispersion minimizing compact finite difference scheme for the 2D Helmholtz equation. J. Comput. Appl. Math. 311, 497–512 (2017)

    Article  MathSciNet  Google Scholar 

  38. Wu, T., Xu, R.: An optimal compact sixth-order finite difference scheme for the Helmholtz equation. Comput. Math. Appl. 75(7), 2520–2537 (2018)

    Article  MathSciNet  Google Scholar 

  39. Zeng, Y., He, J., Liu, Q.: The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media. Geophysics 66(4), 1258–1266 (2001)

    Article  Google Scholar 

Download references

Funding

The work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through the individual Discovery Grant (RGPIN-2019-04830). The first author is also supported by the Alberta Innovates Graduate Student Scholarship that he received during his PhD study.

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Authors and Affiliations

  1. Department of Mathematics & Statistics, University of Calgary, AB, T2N 1N4, Calgary, Canada

    Hatef Dastour & Wenyuan Liao

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  1. Hatef Dastour
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  2. Wenyuan Liao
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Correspondence to Hatef Dastour.

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Dastour, H., Liao, W. An optimal 13-point finite difference scheme for a 2D Helmholtz equation with a perfectly matched layer boundary condition. Numer Algor 86, 1109–1141 (2021). https://doi.org/10.1007/s11075-020-00926-5

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  • DOI: https://doi.org/10.1007/s11075-020-00926-5

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