Symbolic Manipulation of Flows of Nonlinear Evolution Equations, with Application in the Analysis of Split-Step Time Integrators

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

Abstract

We describe a package realized in the Julia programming language which performs symbolic manipulations applied to nonlinear evolution equations, their flows, and commutators of such objects. This tool was employed to perform contrived computations arising in the analysis of the local error of operator splitting methods. It enabled the proof of the convergence of the basic method and of the asymptotical correctness of a defect-based error estimator. The performance of our package is illustrated on several examples.

Keywords

Nonlinear evolution equations Time integration Splitting methods Symbolic computation Julia language 

References

  1. 1.
  2. 2.
  3. 3.
    The On-line Encyclopedia of Integer Sequences. https://oeis.org/A000081
  4. 4.
    Auzinger, W., Hofstätter, H., Koch, O., Thalhammer, M.: Defect-based local error estimators for splitting methods, with application to Schrödinger equations, part III: the nonlinear case. J. Comput. Appl. Math. 273, 182–204 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration, 2nd edn. Springer, Heidelberg (2006)Google Scholar
  6. 6.
    Hairer, E., Nørsett, S., Wanner, G.: Solving Ordinary Differential Equations I, 2nd edn. Springer, Heidelberg (1993)Google Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

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

Personalised recommendations