Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Stability at Nonconforming Grid Interfaces for a High Order Discretization of the Schrödinger Equation

  • 414 Accesses

  • 12 Citations


In this paper we extend the results from our earlier work on stable boundary closures for the Schrödinger equation using the summation-by-parts-simultaneous approximation term (SBP–SAT) method to include stability and accuracy at nonconforming grid interfaces. Stability at the grid interface is shown by the energy method, and the estimates are generalized to multiple dimensions. The accuracy of the grid interface coupling is investigated using normal mode analysis for operators of 2nd and 4th order formal interior accuracy. We show that full accuracy is retained for the 2nd and 4th order operators. The accuracy results are extended to 6th and 8th order operators by numerical simulations, in which case two orders of accuracy is gained with respect to the lower order approximation close to the interface.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9


  1. 1.

    Berger, M.J., Oliger, J.: Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53, 484–512 (1984)

  2. 2.

    Stout, Q., De Zeeuw, D., Gombosi, T., Groth, C., Marshall, H., Powell, K.: Adaptive blocks: A high-performance data structure. In: Proceedings of the 1997 ACM/IEEE SC97 Conference (1997)

  3. 3.

    Rantakokko, J., Thuné, M.: Parallel structured adaptive mesh refinement. In: Troubec, R. et al. (eds.) Parallel Computing. Springer, London, pp. 147–173 (2009)

  4. 4.

    Ferm, L., Lötstedt, P.: Accurate and stable grid interfaces for finite volume methods. Appl. Numer. Math. 49, 207–224 (2004)

  5. 5.

    Choi, D.-I., Brown, J.D., Imbibira, B., Centrella, J., MacNiece, P.: Interface conditions for wave propagations through mesh refinement boundaries. J. Comput. Phys. 193, 398–425 (2004)

  6. 6.

    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)

  7. 7.

    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)

  8. 8.

    Carpenter, M.H., Nordström, J., Gottlieb, D.: A stable and conservative interface treatment of arbitrary spatial accuracy. J. Comput. Phys. 148, 341–365 (1999)

  9. 9.

    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)

  10. 10.

    Nordström, J., Carpenter, M.H.: Boundary and interface conditions for high-order finite-difference methods applied to the Euler and Navier-Stokes equations. J. Comput. Phys. 148, 621–645 (1999)

  11. 11.

    Nordström, J., Gong, J., van der Weide, E., Svärd, M.: A stable and conservative high order multi-block method for the compressible Navier-Stokes equations. J. Comput. Phys. 228, 9020–9035 (2009)

  12. 12.

    Reula, O.: Numerical treatment of interfaces in quantum dynamics (2011). arXiv:1103.5448v1 [quant-ph]

  13. 13.

    Carpenter, M.H., Nordström, J., Gottlieb, D.: Revisiting and extending interface penalties for multi-domain summation-by-parts operators. J. Sci. Comput. 45, 118–150 (2010)

  14. 14.

    Lindström, J., Nordström, J.: A stable and high-order accurate conjugate heat transfer problem. J. Comput. Phys. 229, 5440–5456 (2010)

  15. 15.

    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)

  16. 16.

    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)

  17. 17.

    Nissen, A., Kreiss, G., Gerritsen, M.: High order stable finite difference methods for the Schrödinger equation. Technical report nr 2011-014, Department of Information Technology, Uppsala University (2011)

  18. 18.

    Svärd, M., Nordström, J.: On the order of accuracy for difference approximations of initial-boundary value problems. J. Comput. Phys. 218, 333–352 (2006)

  19. 19.

    Gustafsson, B., Kreiss, H.-O., Oliger, J.: Time-Dependent Problems and Difference Methods. Wiley, New York (1995)

  20. 20.

    Mattsson, K., Nordström, J.: Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys. 199, 503–540 (2004)

  21. 21.

    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 (2008)

Download references


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.

Author information

Correspondence to A. Nissen.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Nissen, A., Kreiss, G. & Gerritsen, M. Stability at Nonconforming Grid Interfaces for a High Order Discretization of the Schrödinger Equation. J Sci Comput 53, 528–551 (2012).

Download citation


  • Schrödinger equation
  • Finite difference methods
  • Summation-by-parts operator
  • Stability
  • Grid interface