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Setup of Order Conditions for Splitting Methods

  • Winfried Auzinger
  • Wolfgang Herfort
  • Harald Hofstätter
  • Othmar Koch
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9890)

Abstract

For operator splitting methods, an approach based on Taylor expansion and the particular structure of its leading term as an element of a free Lie algebra is used for the setup of a system of order conditions. Along with a brief review of the underlying theoretical background, we discuss the implementation of the resulting algorithm in computer algebra, in particular using Maple 18 (Maple is a trademark of MapleSoft\(^\mathrm{TM}\).). A parallel version of such a code is described, and its performance on a computational node with 16 threads is documented.

Keywords

Evolution equations Splitting methods Order conditions Local error Taylor expansion Parallel processing 

Notes

Acknowledgements

This work was supported by the Austrian Science Fund (FWF) under grant P24157-N13, and by the Vienna Science and Technology Fund (WWTF) under grant MA-14-002. Computational results based on the ideas in this work have been achieved in part using the Vienna Scientific Cluster (VSC).

References

  1. 1.
    Auzinger, W., Herfort, W.: Local error structures and order conditions in terms of Lie elements for exponential splitting schemes. Opuscula Math. 34, 243–255 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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. To appear in BIT Numer. MathGoogle Scholar
  3. 3.
    Auzinger, W., Koch, O.: Coefficients of various splitting methods. www.asc.tuwien.ac.at/ winfried/splitting/
  4. 4.
    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
  5. 5.
    Blanes, S., Casas, F., Farrés, A., Laskar, J., Makazaga, J., Murua, A.: New families of symplectic splitting methods for numerical integration in dynamical astronomy. Appl. Numer. Math. 68, 58–72 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bokut, L., Sbitneva, L., Shestakov, I.: Lyndon-Shirshov words, Gröbner-Shirshov bases, and free Lie algebras. In: Non-Associative Algebra and Its Applications, chap 3. Chapman & Hall / CRC, Boca Raton (2006)Google Scholar
  7. 7.
    Duval, J.P.: Géneration d’une section des classes de conjugaison et arbre des mots de Lyndon de longueur bornée. Theoret. Comput. Sci. 60, 255–283 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration, 2nd edn. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  9. 9.
    Ketcheson, D., MacDonald, C., Ruuth, S.: Spatially partitioned embedded Runge-Kutta methods. SIAM J. Numer. Anal. 51, 2887–2910 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Winfried Auzinger
    • 1
  • Wolfgang Herfort
    • 1
  • Harald Hofstätter
    • 1
  • Othmar Koch
    • 2
  1. 1.Technische Universität WienViennaAustria
  2. 2.Universität WienViennaAustria

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