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WKB-method for the 1D Schrödinger equation in the semi-classical limit: enhanced phase treatment

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Abstract

This paper is concerned with the efficient numerical computation of solutions to the 1D stationary Schrödinger equation in the semiclassical limit in the highly oscillatory regime. A previous approach to this problem based on explicitly incorporating the leading terms of the WKB approximation is enhanced in two ways: first a refined error analysis for the method is presented for a not explicitly known WKB phase, and secondly the phase and its derivatives will be computed with spectral methods. The efficiency of the approach is illustrated for several examples.

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References

  1. Arnold, A., Ben Abdallah, N., Negulescu, C.: WKB-based schemes for the oscillatory 1D Schrödinger equation in the semi-classical limit. SIAM J. Numer. Anal. 49(4), 1436–1460 (2011)

    Article  MathSciNet  Google Scholar 

  2. Arnold, A., Döpfner, K.: Stationary Schrödinger equation in the semi-classical limit: WKB-based scheme coupled to a turning point. Calcolo 57, 3 (2020). https://doi.org/10.1007/s10092-019-0349-9

  3. Arnold, A., Negulescu, C.: Stationary Schrödinger equation in the semi-classical limit: numerical coupling of oscillatory and evanescent regions. Numer. Math. 138(2), 501–536 (2018)

    Article  MathSciNet  Google Scholar 

  4. Berrut, J.-P., Trefethen, L.N.: Barycentric Lagrange Interpolation. SIAM Review 46(3), 501–517 (2004)

    Article  MathSciNet  Google Scholar 

  5. Chawla, M.M.: Error Estimates for the Clenshaw-Curtis Quadrature. Math. Comp. 22(103), 651–656 (1968)

    Article  MathSciNet  Google Scholar 

  6. Clenshaw, C.W., Curtis, A.R.: A method for numerical integration on an automatic computer. Numerische Mathematik 2(1), 197–205 (1960)

    Article  MathSciNet  Google Scholar 

  7. Cohen, D., Hairer, E., Lubich, C.: Modulated Fourier Expansions of Highly Oscillatory Differential Equations. Found. Comput. Math. 3, 327–345 (2003)

    Article  MathSciNet  Google Scholar 

  8. Degond, P., Gallego, S., Méhats, F.: An asymptotic preserving scheme for the Schrödinger equation in the semiclassical limit. C.R. Acad. Sci. Paris, Ser. I 345(9), 531–536 (2007)

    Article  Google Scholar 

  9. E, W., Engquist, B., Li, X., Ren, W., Vanden-Eijnden, E.: Heterogeneous multiscale methods: a review, Commun. Comput. Phys. 2(3), 367–450 (2007)

  10. Geier, J.: Efficient integrators for linear highly oscillatory ODEs based on asymptotic expansions, PhD-dissertation at TU Wien, (2011)

  11. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer-Verlag, Berlin Heidelberg (2006)

    MATH  Google Scholar 

  12. Handley, W.J., Lasenby, A.N., Hobson, M.P.: The Runge-Kutta-Wentzel-Kramers-Brillouin method, preprint, (2016). arXiv:1612.02288

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

    Article  MathSciNet  Google Scholar 

  14. Ihlenburg, F., Babuška, I.: Finite element solution of the Helmholtz equation with high wave number. II. The \(h\)-\(p\) version of the FEM. SIAM J. Numer. Anal. 34(1), 315–358 (1997)

    Article  MathSciNet  Google Scholar 

  15. Iserles, A.: On the Global Error of Discretization Methods for Highly-Oscillatory Ordinary Differential Equations. BIT 42(3), 561–599 (2002)

    Article  MathSciNet  Google Scholar 

  16. Iserles, A., Munthe-Kaas, H.Z., Nørsett, S.P., Zanna, A.: Lie-group methods. Acta Numerica 9, 215–365 (2000)

    Article  MathSciNet  Google Scholar 

  17. Iserles, A., Nørsett, S.P., Olver, S.: Highly oscillatory quadrature: The story so far. In: Bermudez de Castro, A. (ed.) Proceeding of ENuMath, Santiago de Compostella (2006), pp. 97–118. Springer Verlag, (2006)

  18. Jahnke, T.: Long-time-step integrators for almost-adiabatic quantum dynamics. SIAM J. Sci. Comp. 25, 2145–2164 (2004)

    Article  MathSciNet  Google Scholar 

  19. Jahnke, T., Lubich, C.: Numerical integrators for quantum dynamics close to the adiabatic limit. Numerische Mathematik 94, 289–314 (2003)

    Article  MathSciNet  Google Scholar 

  20. Landau, L.D., Lifschitz, E.M.: Quantenmechanik. Akademie-Verlag, Berlin (1985)

    Google Scholar 

  21. Lent, C.S., Kirkner, D.J.: The Quantum Transmitting Boundary Method. J. Appl. Phys. 67, 6353–6359 (1990)

    Article  Google Scholar 

  22. Lorenz, K., Jahnke, T., Lubich, C.: Adiabatic integrators for highly oscillatory second-order linear differential equations with time-varying eigendecomposition. BIT 45(1), 91–115 (2005)

    Article  MathSciNet  Google Scholar 

  23. Mennemann, J.-F., Jüngel, A., Kosina, H.: Transient Schrödinger-Poisson simulations of a high-frequency resonant tunneling diode oscillator. J. Computat. Phys. 239, 187–205 (2013)

    Article  Google Scholar 

  24. Moan, P.C., Niesen, J.: Convergence of the Magnus Series. Found. Comput. Math. 8, 291–301 (2008)

    Article  MathSciNet  Google Scholar 

  25. Negulescu, C.: Numerical analysis of a multiscale finite element scheme for the resolution of the stationary Schrödinger equation. Numerische Mathematik 108(4), 625–652 (2008)

    Article  MathSciNet  Google Scholar 

  26. Olver, S.: Moment-free numerical integration of highly oscillatory functions. IMA J. Numer. Analy. 26, 213–227 (2006)

    Article  MathSciNet  Google Scholar 

  27. Sun, J.P., Haddad, G.I., Mazumder, P., Schulman, J.N.: Resonant Tunneling Diodes: Models and Properties. Proc. of the IEEE 86(4), 641–661 (1998)

    Article  Google Scholar 

  28. Trefethen, L.N.: Spectral Methods in MATLAB, vol. 10 of Software, Environments, and Tools, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (2000)

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Authors and Affiliations

  1. Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8-10, 1040, Wien, Austria

    Anton Arnold & Bernhard Ujvari

  2. Institut de Mathématiques de Bourgogne, Université de Bourgogne-Franche-Comté, 9 Avenue Alain Savary, Dijon, France

    Christian Klein

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  1. Anton Arnold
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  2. Christian Klein
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  3. Bernhard Ujvari
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Correspondence to Christian Klein.

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Communicated by Mechthild Thalhammer.

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The first author (AA) was supported by the FWF-doctoral school “Dissipation and dispersion in non-linear partial differential equations”, the bi-national FWF-project I3538-N32, and a sponsorship by Clear Sky Ventures. This work was supported by the ANR-FWF project ANuI. CK thanks for support by the isite BFC project NAANoD, the ANR-17-EURE-0002 EIPHI and by the European Union Horizon 2020 research and innovation program under the Marie Sklodowska-Curie RISE 2017 Grant Agreement No. 778010 IPaDEGAN.

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Arnold, A., Klein, C. & Ujvari, B. WKB-method for the 1D Schrödinger equation in the semi-classical limit: enhanced phase treatment. Bit Numer Math 62, 1–22 (2022). https://doi.org/10.1007/s10543-021-00868-x

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