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A Second-Order Fast Huygens Sweeping Method for Time-Dependent Schrödinger Equations with Perfectly Matched Layers

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Abstract

The time-dependent Schrödinger equation is generally challenging to solve numerically since the problem is often defined in the infinite domain and the wavefunction is oscillatory. In order to tackle the problem, we extend the fast Huygens sweeping method (FHSM) in Leung et al. (Methods Appl Anal 21(1):31–66, 2014) by combining it with the perfectly matched layer (PML) method and Strang operator splitting techniques. Firstly, the PML approach is applied to limit the infinite domain to a bounded sub-domain of interest, where a modified Schrödinger equation is derived. Secondly, the modified Schrödinger equation is solved by the FHSM in the bounded sub-domain, where a Huygens’ principle based short-time propagator in the form of an integral with retarded Green’s functions is used to propagate the wavefunction, and the retarded Green’s function is approximated by the semi-classical limit approximations with the phase and amplitude terms obtained as solutions to the eikonal and transport equations, respectively. In the case of nonlinear Schrödinger equations, the Strang operator splitting approach will be utilized to transfer the modified equation into sub-problems such that one of them has analytic solutions and the other one is still an equation of Schrödinger type whose wavefunction can be propagated by the FHSM. Finally, an improved randomized procedure with the pivoted QR factorization is designed to construct appropriate low-rank approximations such that the resulting approximated integral can be evaluated with the fast Fourier transform. Numerical examples are presented to demonstrate the method.

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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Antoine, X., Arnold, A., Besse, C., Ehrhardt, M., Schaedle, A.: A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations. Commun. Comput. Phys. 4, 729–796 (2008)

    MathSciNet  MATH  Google Scholar 

  2. Antoine, X., Bao, W., Besse, C.: Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations. Comput. Phys. Commun. 184(12), 2621–2633 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Babuska, I., Ihlenburg, F., Strouboulis, T., Gangaraj, S.K.: A posteriori error estimation for finite element solutions of Helmholtz’ equation. Part i: the quality of local indicators and estimators. Int. J. Numer. Methods Eng. 40(18), 3443–3462 (1997)

    MATH  Google Scholar 

  4. 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 

  5. Baker, B., Copson, E.T.: The Mathematical Theory of Huygens’ Principle. AMS Chelsea Publishing, New York (1987)

    MATH  Google Scholar 

  6. 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)

    MathSciNet  MATH  Google Scholar 

  7. Bao, W., Jin, S., Markowich, P.: Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes. SIAM J. Sci. Comput. 25(1), 27–64 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Bao, W., Jin, S., Markowich, P.A.: On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime. J. Comput. Phys. 175(2), 487–524 (2002)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  10. Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover, New York (2001)

    MATH  Google Scholar 

  11. Brack, M., Bhaduri, R.K.: Semiclassical Physics. Addison Wesley, Boston (1997)

    MATH  Google Scholar 

  12. Canés, E., Demanet, L., Ying, L.: A fast butterfly algorithm for the computation of Fourier integral operators. Multiscale Model. Simul. 7(4), 1727–1750 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Chand, P., Hoekstra, J.: A review of the semi-classical WKB approximation and its usefulness in the study of quantum systems. In: Proc. of IEEE Semiconductor Advances for Future Electronics, pp. 13–19 (2001)

  14. Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)

    MathSciNet  MATH  Google Scholar 

  15. Crandall, M.G., Lions, P.-L.: Viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 277(1), 1–42 (1983)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Engquist, B., Ying, L.: A fast directional algorithm for high frequency acoustic scattering in two dimensions. Commun. Math. Sci. 7(2), 327–345 (2009)

    MathSciNet  MATH  Google Scholar 

  18. Fomel, S., Ying, L., Song, X.: Seismic wave extrapolation using Lowrank symbol approximation. Geophys. Prospect. 61(3), 526–536 (2013)

    Google Scholar 

  19. Fujiwara, D.: A construction of the fundamental solution for the Schrödinger equations. Proc. Jpn. Acad. Ser. A Math. Sci. 55(1), 10–14 (1979)

    MATH  Google Scholar 

  20. Goreinov, S.A., Zamarashkin, N.L., Tyrtyshnikov, E.E.: Pseudo-skeleton approximations by matrices of maximal volume. Math. Notes 62(4), 515–519 (1997)

    MathSciNet  MATH  Google Scholar 

  21. Goreinov, S.A., Tyrtyshnikov, E.E., Zamarashkin, N.L.: A theory of pseudoskeleton approximations. Linear Algebra Appl. 261(1), 1–21 (1997)

    MathSciNet  MATH  Google Scholar 

  22. Jin, S., Markowich, P., Sparber, C.: Mathematical and computational methods for semiclassical Schrödinger equations. Acta Numerica 20, 121–209 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Keller, J.B.: Semiclassical mechanics. SIAM Rev. 27, 485–504 (1985)

    MathSciNet  MATH  Google Scholar 

  24. Kitada, H., Kumano-go, H.: A family of Fourier integral operators and the fundamental solution for a Schrödinger equation. Osaka J. Math. 18(2), 291–360 (1981)

    MathSciNet  MATH  Google Scholar 

  25. Leung, S., Qian, J., Serna, S.: Fast Huygens sweeping methods for Schrödinger equations in the semi-classical regime. Methods Appl. Anal. 21(1), 31–66 (2014)

    MathSciNet  MATH  Google Scholar 

  26. Lu, W., Qian, J., Burridge, R.: Extending Babich’s ansatz for point-source Maxwell’s equations using Hadamard’s method. Multiscale Model. Simul. 16(2), 727–751 (2018)

    MathSciNet  MATH  Google Scholar 

  27. Luo, S.: Fast Huygens sweeping methods for time-dependent Schrödinger equation with perfectly matched layers. SIAM J. Sci. Comput. 41(2), A877–A899 (2019)

    MATH  Google Scholar 

  28. 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)

    MathSciNet  MATH  Google Scholar 

  29. Luo, S., Qian, J., Burridge, R.: High-order factorization based high-order hybrid fast sweeping methods for point-source Eikonal equations. SIAM J. Numer. Anal. 52(1), 23–44 (2014)

    MathSciNet  MATH  Google Scholar 

  30. Martinez, A.: An Introduction to Semiclassical and Microlocal Analysis. Springer, New York (2002)

    MATH  Google Scholar 

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

    Google Scholar 

  32. 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)

    MathSciNet  MATH  Google Scholar 

  33. 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)

    Google Scholar 

  34. Runge, E., Gross, E.K.U.: Density-functional theory for time-dependent systems. Phys. Rev. Lett. 52, 997–1000 (1984)

    Google Scholar 

  35. Schrödinger, E.: An undulatory theory of the mechanics of atoms and molecules. Phys. Rev. 28, 1049–1070 (1926)

    Google Scholar 

  36. Schulman, L.S.: Techniques and Applications of Path Integration. Wiley, New York (1981)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  38. Zagoskin, A.: Quantum Theory of Many-Body Systems: Techniques and Applications, 2nd edn. Springer Publishing Company, Berlin (2014)

    MATH  Google Scholar 

  39. Zheng, C.: A perfectly matched layer approach to the nonlinear Schrödinger wave equations. J. Comput. Phys. 227(1), 537–556 (2007)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

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

    Yijin Gao, Jay Mayfield & Songting Luo

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  1. Yijin Gao
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  2. Jay Mayfield
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  3. Songting Luo
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Correspondence to Songting Luo.

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The codes generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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This work was partially supported by NSF DMS 1418908 and 1719907.

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Gao, Y., Mayfield, J. & Luo, S. A Second-Order Fast Huygens Sweeping Method for Time-Dependent Schrödinger Equations with Perfectly Matched Layers. J Sci Comput 88, 49 (2021). https://doi.org/10.1007/s10915-021-01560-6

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  • DOI: https://doi.org/10.1007/s10915-021-01560-6

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