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

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Abstract

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.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

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

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. Department of Information Technology, Uppsala University, Uppsala, Sweden

    G. Kreiss

  2. Department of Energy Resources Engineering, Stanford University, Stanford, CA, USA

    A. Nissen & M. Gerritsen

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  1. A. Nissen
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  2. G. Kreiss
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  3. M. Gerritsen
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Correspondence to A. Nissen.

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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). https://doi.org/10.1007/s10915-012-9586-7

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  • DOI: https://doi.org/10.1007/s10915-012-9586-7

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