Journal of Scientific Computing

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

High Order Stable Finite Difference Methods for the Schrödinger Equation



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.


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



The authors would like to thank Katharina Kormann for help with the time-stepping method. This work has been partially financed by Anna Maria Lundins stipendiefond and the Swedish research council. The computations were performed on resources provided by SNIC through Uppsala Multidisciplinary Center for Advanced Computational Science (UPPMAX) under Project p2005005.


  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

Personalised recommendations