Journal of Scientific Computing

, Volume 55, Issue 1, pp 173–199 | Cite as

High Order Stable Finite Difference Methods for the Schrödinger Equation

Article

Abstract

In this paper we extend the Summation-by-parts-simultaneous approximation term (SBP-SAT) technique to the Schrödinger equation. Stability estimates are derived and the accuracy of numerical approximations of interior order 2m, m=1,2,3, are analyzed in the case of Dirichlet boundary conditions. We show that a boundary closure of the numerical approximations of order m lead to global accuracy of order m+2. The results are supported by numerical simulations.

Keywords

Schrödinger equation High-order finite difference methods Summation-by-parts operators Numerical stability 

References

  1. 1.
    Born, M., Oppenheimer, R.: Zur Quantentheorie der Molekeln. Ann. Phys. 84, 457–484 (1927) MATHCrossRefGoogle Scholar
  2. 2.
    Kosloff, D., Kosloff, R.: A Fourier method solution for the time dependent Schrödinger equation as a tool in molecular dynamics. J. Comput. Phys. 52, 35–53 (1983) MATHCrossRefGoogle Scholar
  3. 3.
    Carpenter, M.H., Gottlieb, D., Abarbanel, S.: Time-stable boundary conditions for finite difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes. J. Comput. Phys. 111, 220–236 (1994) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Kreiss, H.-O., Scherer, G.: Finite element and finite difference methods for hyperbolic partial differential equations. In: Mathematical Aspects of Finite Elements in Partial Differential Equations. Academic Press, New York (1974) Google Scholar
  5. 5.
    Strand, B.: Summation by parts for finite difference approximations for d/dx. J. Comput. Phys. 110, 47–67 (1994) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Carpenter, M.H., Nordström, J., Gottlieb, D.: A stable and conservative interface treatment of arbitrary spatial accuracy. J. Comput. Phys. 148, 341–365 (1999) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Mattsson, K., Carpenter, M.H.: Stable and accurate interpolation operators for high-order multiblock finite difference methods. SIAM J. Sci. Comput. 32(4), 2298–2320 (2010) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Kramer, R.M.J., Pantano, C., Pullin, D.I.: A class of energy stable, high-order finite-difference interface schemes suitable for adaptive mesh refinement of hyperbolic problems. J. Comput. Phys. 226, 1458–1484 (2007) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Kramer, R.M.J., Pantano, C., Pullin, D.I.: Nondissipative and energy-stable high-order finite-difference interface schemes for 2-D patch-refined grids. J. Comput. Phys. 228, 5280–5297 (2009) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Reula, O.: Numerical treatment of interfaces in quantum dynamics. March (2011) Google Scholar
  11. 11.
    Nissen, A., Kreiss, G., Gerritsen, M.: Stability at nonconforming grid interfaces for a high order discretization of the Schrödinger equation. J. Sci. Comput. (2012). doi:10.1007/s10915-012-9586-7 MathSciNetMATHGoogle Scholar
  12. 12.
    Gustafsson, B.: The convergence rate for difference approximations to mixed initial boundary value problems. Math. Comput. 29, 396–406 (1975) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Gustafsson, B.: The convergence rate for difference approximations to general mixed initial boundary value problems. SIAM J. Numer. Anal. 18, 179–190 (1981) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Svärd, M., Nordström, J.: On the order of accuracy for difference approximations of initial-boundary value problems. J. Comput. Phys. 218, 333–352 (2006) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Mattsson, K., Ham, F., Iaccarino, G.: Stable boundary treatment for the wave equation on second-order form. J. Sci. Comput. 41, 366–383 (2009) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Gustafsson, B., Kreiss, H.-O., Oliger, J.: Time-Dependent Problems and Difference Methods. Wiley, New York (1995) MATHGoogle Scholar
  17. 17.
    Mattsson, K., Nordström, J.: Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys. 199, 503–540 (2004) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Baumann, C.E., Oden, J.T.: A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Eng. 175, 311–341 (1999) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Kormann, K., Holmgren, S., Karlsson, H.O.: Accurate time propagation for the Schrödinger equation with an explicitly time-dependent Hamiltonian. J. Chem. Phys. 128(184101), 1–11 (2008) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Energy Resources EngineeringStanford UniversityStanfordUSA
  2. 2.Department of Information TechnologyUppsala UniversityUppsalaSweden

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