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

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Computer Algebra in Scientific Computing (CASC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9890))

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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.

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Notes

  1. 1.

    Likewise for objects of type NonAutonomousFunction.

References

  1. http://julialang.org

  2. https://github.com/HaraldHofstaetter/Flows.jl

  3. The On-line Encyclopedia of Integer Sequences. https://oeis.org/A000081

  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)

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  5. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration, 2nd edn. Springer, Heidelberg (2006)

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  6. Hairer, E., Nørsett, S., Wanner, G.: Solving Ordinary Differential Equations I, 2nd edn. Springer, Heidelberg (1993)

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Acknowledgments

This work was supported in part by the projects P24157-N13 of the Austrian Science Fund (FWF) and MA14-002 of the Vienna Science and Technology Fund (WWTF).

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Authors and Affiliations

  1. Technische Universität Wien, Vienna, Austria

    Winfried Auzinger & Harald Hofstätter

  2. Universität Wien, Vienna, Austria

    Othmar Koch

Authors
  1. Winfried Auzinger
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  2. Harald Hofstätter
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  3. Othmar Koch
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Corresponding author

Correspondence to Harald Hofstätter .

Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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© 2016 Springer International Publishing AG

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Auzinger, W., Hofstätter, H., Koch, O. (2016). Symbolic Manipulation of Flows of Nonlinear Evolution Equations, with Application in the Analysis of Split-Step Time Integrators. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science(), vol 9890. Springer, Cham. https://doi.org/10.1007/978-3-319-45641-6_4

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  • DOI: https://doi.org/10.1007/978-3-319-45641-6_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45640-9

  • Online ISBN: 978-3-319-45641-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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