Journal of Scientific Computing

, Volume 65, Issue 2, pp 486–511 | Cite as

Stable Difference Methods for Block-Oriented Adaptive Grids

  • Anna NissenEmail author
  • Katharina Kormann
  • Magnus Grandin
  • Kristoffer Virta


In this paper, we present a block-oriented scheme for adaptive mesh refinement based on summation-by-parts (SBP) finite difference methods and simultaneous-approximation-term (SAT) interface treatment. Since the order of accuracy at SBP–SAT grid interfaces is lower compared to that of the interior stencils, we strive at using the interior stencils across block-boundaries whenever possible. We devise a stable treatment of SBP-FD junction points, i.e. points where interfaces with different boundary treatment meet. This leads to stable discretizations for more flexible grid configurations within the SBP–SAT framework, with a reduced number of SBP–SAT interfaces. Both first and second derivatives are considered in the analysis. Even though the stencil order is locally reduced close to numerical interfaces and corner points, numerical simulations show that the locally reduced accuracy does not severely reduce the accuracy of the time propagated numerical solution. Moreover, we explain how to organize the grid and how to automatically adapt the mesh, aiming at problems of many variables. Examples of adaptive grids are demonstrated for the simulation of the time-dependent Schrödinger equation and for the advection equation.


Summation-by-parts Simultaneous-approximating-term  Block-structured grid Adaptive mesh refinement Time-dependent Schrödinger equation Advection equation 



The authors would like to thank Sverker Holmgren and Gunilla Kreiss for valuable insight and discussions. The design of the interpolation operators is based on a Maple sheet by Ken Mattsson. The simulations were performed on resources provided by SNIC-UPPMAX under Projects p2003013 and p2005005.


  1. 1.
    Appelö, D., Petersson, N.A.: A stable finite difference method for the elastic wave equation on complex geometries with free surfaces. Commun. Comput. Phys. 5, 84–107 (2008)Google Scholar
  2. 2.
    Berger, M.J., Colella, P.: Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 82, 64–84 (1989)zbMATHCrossRefGoogle Scholar
  3. 3.
    Berger, M.J., Oliger, J.: Adaptive mesh refinement for hyperbolic partial differential equation. J. Comput. Phys. 53, 484–512 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    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)Google Scholar
  5. 5.
    Dreher, J., Grauer, R.: Racoon: a parallel mesh-adaptive framework for hyperbolic conservation laws. Parallel Comput. 31(89), 913–932 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ferm, L., Hellander, A., Lötstedt, P.: An adaptive algorithm for simulation of stochastic reaction–diffusion processes. J. Comput. Phys. 229, 343–360 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Gustafsson, B.: The convergence rate for difference approximations to mixed initial boundary value problems. Math. Comput. 29, 396–406 (1975)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Gustafsson, B.: The convergence rate for difference approximations to general mixed initial boundary value problems. SIAM J. Numer. Anal. 18, 179–190 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Kormann, K.: A time-space adaptive method for the Schrödinger equation. Tech. Rep. 2012–023. Department of Information Technology, Uppsala University (2012)Google Scholar
  10. 10.
    Kormann, K., Holmgren, S., Karlsson, H.O.: Global error control of the time-propagation for the Schrödinger equation with a time-dependent Hamiltonian. J. Comput. Sci. 2, 178–187 (2011)CrossRefGoogle Scholar
  11. 11.
    Kormann, K., Nissen, A.: Error control for simulations of a dissociative quantum system. In: Kreiss, G., Lötstedt, P., Målqvist, A., Neytcheva, M. (eds.) Numerical Mathematics and Advanced Applications 2009, pp. 523–531. Springer, Berlin Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Kozdon, J.E., Dunham, E.M., Nordström, J.: Simulation of dynamic earthquake ruptures in complex geometries using high-order finite difference methods. J. Sci. Comput. 55, 92–124 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Kreiss, H.-O., Scherer, G.: Finite element and finite difference methods for hyperbolic partial differential equations. In: de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 195–212. Academic Press, New York (1974)Google Scholar
  15. 15.
    Lindström, J., Nordström, J.: A stable and high-order accurate conjugate heat transfer problem. J. Comput. Phys. 229, 5440–5456 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    MacNeice, P., Olson, K.M., Mobarry, C., de Fainchtein, R., Packer, C.: PARAMESH: a parallel adaptive mesh refinement community toolkit. Comput. Phys. Commun. 126, 330–354 (2000)zbMATHCrossRefGoogle Scholar
  17. 17.
    Mattsson, K., Carpenter, M.H.: Stable and accurate interpolation operators for high-order multi-block finite-difference methods. SIAM J. Sci. Comput. 32, 2298–2320 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Mattsson, K., Nordström, J.: Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys. 199, 503–540 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Meyer, H.-D., Munthe, U., Cederbaum, L.S.: The multi-configurational time-dependent Hartree approach. Chem. Phys. Lett. 165, 73–78 (1990)CrossRefGoogle Scholar
  20. 20.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Nissen, A., Kreiss, G., Gerritsen, M.: High order stable finite difference methods for the Schrödinger equation. J. Sci. Comput. 55, 173–199 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Rantakokko, J., Thuné, M.: Parallel structured adaptive mesh refinement. In: Trobec, R., Vajteric, M., Zinterhof, P. (eds.) Parallel Computing, pp. 147–173. Springer, London (2009)CrossRefGoogle Scholar
  23. 23.
    Strand, B.: Summation by parts for finite difference approximations for \(d/dx\). J. Comput. Phys. 110, 47–67 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Svärd, M., Nordström, J.: On the order of accuracy for difference approximations of initial-boundary value problems. J. Comput. Phys. 218, 333–352 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Svärd, M., Nordström, J.: A stable high-order finite difference scheme for the compressible Navier–Stokes equations: no-slip wall boundary conditions. J. Comput. Phys. 227, 4805–4824 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Tannor, D.J.: Introduction to Quantum Mechanics: A Time-Dependent Perspective. University Science Book, Mill Valley (2007)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Anna Nissen
    • 1
    Email author
  • Katharina Kormann
    • 2
  • Magnus Grandin
    • 3
  • Kristoffer Virta
    • 3
  1. 1.Department of MathematicsUniversity of BergenBergenNorway
  2. 2.Zentrum MathematikTechnische Universität MünchenGarchingGermany
  3. 3.Department of Information TechnologyUppsala UniversityUppsalaSweden

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