Advertisement

Numerical Algorithms

, Volume 75, Issue 1, pp 261–283 | Cite as

Adaptive high-order splitting methods for systems of nonlinear evolution equations with periodic boundary conditions

  • Winfried Auzinger
  • Othmar Koch
  • Michael Quell
Open Access
Original Paper

Abstract

We assess the applicability and efficiency of time-adaptive high-order splitting methods applied for the numerical solution of (systems of) nonlinear parabolic problems under periodic boundary conditions. We discuss in particular several applications generating intricate patterns and displaying nonsmooth solution dynamics. First, we give a general error analysis for splitting methods for parabolic problems under periodic boundary conditions and derive the necessary smoothness requirements on the exact solution in particular for the Gray–Scott equation and the Van der Pol equation. Numerical examples demonstrate the convergence of the methods and serve to compare the efficiency of different time-adaptive splitting schemes and of splitting into either two or three operators, based on appropriately constructed a posteriori local error estimators.

Keywords

Nonlinear evolution equations Splitting methods Adaptive time integration Local error Convergence 

Mathematics Subject Classification (2010)

65J10 65L05 65M12 65M15 

References

  1. 1.
    Söderlind, G., Wang, L.: Adaptive time-stepping and computational stability. J. Comput. Appl. Math. 185, 225–243 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. Springer-Verlag, Berlin–Heidelberg–New York (2002)CrossRefzbMATHGoogle Scholar
  3. 3.
    Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Descombes, S., Duarte, M., Dumont, T., Louvet, V., Massot, M.: Adaptive time splitting method for multi-scale evolutionary partial differential equations. Confluentes Math. 03, 413–443 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Jahnke, T., Lubich, C.: Error bounds for exponential operator splittings. BIT 40, 735–744 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Thalhammer, M.: High-order exponential operator splitting methods for time-dependent Schrödinger equations. SIAM J. Numer. Anal. 46(4), 2022–2038 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Lubich, C.: On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations. Math. Comp. 77, 2141–2153 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Koch, O., Neuhauser, C., Thalhammer, M.: Error analysis of high-order splitting methods for nonlinear evolutionary Schrödinger equations and application to the MCTDHF equations in electron dynamics. M2AN Math. Model. Numer. Anal. 47, 1265–1284 (2013)CrossRefzbMATHGoogle Scholar
  9. 9.
    Blanes, S., Casas, F., Chartier, P., Murua, A.: Optimized high-order splitting methods for some classes of parabolic equations. Math. Comp. 82, 1559–1576 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hansen, E., Ostermann, A.: Exponential splitting for unbounded operators. Math. Comp. 78, 1485–1496 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Auzinger, W., Koch, O., Thalhammer, M.: Defect-based local error estimators for splitting methods, with application to Schrödinger equations, Part II: Higher-order methods for linear problems. J. Comput. Appl. Math. 255, 384–403 (2013)CrossRefzbMATHGoogle Scholar
  12. 12.
    Koch, O., Neuhauser, C., Thalhammer, M.: Embedded split-step formulae for the time integration of nonlinear evolution equations. Appl. Numer. Math. 63, 14–24 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Auzinger, W., Koch, O., Thalhammer, M.: Defect-based local error estimators for high-order splitting methods involving three linear operators. Numer. Algorithms 70, 61–91 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Auzinger, W., Hofstätter, H., Ketcheson, D., Koch, O.: Practical splitting methods for the adaptive integration of nonlinear evolution equations. Part I: construction of optimized schemes and pairs of schemes, BIT Numer. Math., published online 28 July 2016Google Scholar
  15. 15.
    Auzinger, W., Koch, O.: Coefficients of various splitting methods, http://www.asc.tuwien.ac.at/~winfried/splitting/
  16. 16.
    Hairer, E., Nørsett, S., Wanner, G.: Solving ordinary differential equations I. Springer-Verlag, Berlin–Heidelberg–New York (1987)CrossRefzbMATHGoogle Scholar
  17. 17.
    Press, W., Flannery, B., Teukolsky, S., Vetterling, W.: Numerical recipes in C — the art of scientific computing. Cambridge University Press, Cambridge (1988)zbMATHGoogle Scholar
  18. 18.
    Gray, P., Scott, S.: Chemical waves and instabilities. Clarendon, Oxford (1990)Google Scholar
  19. 19.
    Robinson, J.: Infinite-dimensional dynamical systems. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  20. 20.
    Katznelson, Y.: An introduction to harmonic analysis. Dover Publications, Inc., New York (1968)zbMATHGoogle Scholar
  21. 21.
    Rudin, W.: Real and complex analysis, 3rd edn. McGraw-Hill (1987)Google Scholar

Copyright information

© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institut für Analysis und Scientific ComputingTechnische Universität WienWienAustria
  2. 2.Fakultät für MathematikUniversität WienWienAustria

Personalised recommendations