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Iterative Implicit Methods for Solving Nonlinear Dynamical Systems: Application of the Levitron

  • Jürgen GeiserEmail author
  • Karl Felix Lüskow
  • Ralf Schneider
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9045)

Abstract

In this paper we apply modified implicit methods for nonlinear dynamical systems related to constrained and non-separable Hamiltonian problems. The application of well-known standard Runge-Kutta integrator methods based on splitting schemes failed, while the energy conservation is no longer guaranteed. We propose a novel class of iterative implicit method that resolves the nonlinearity and achieve an asymptotic symplectic behavior. In comparison to explicit symplectic methods we achieve more accurate results for 5–10 iterations for only double computational time.

Keywords

Semi-implicit integrators Levitron problem Iterative Euler method Crank-Nicolson methods Symplectic splitting Long time computations 

References

  1. 1.
    Dullin, H.R.: Poisson integrator for symmetric rigid bodies. Regul. Chaotic Dyn. 9, 255–264 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Fenga, Q.-D., Jiaoa, Y.-D., Tanga, Y.-F.: Conjugate symplecticity of second-order linear multi-step methods. J. Comput. Appl. Math. 203, 6–14 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Geiser, J.: Nonlinear extension of multiproduct expansion schemes and applications to rigid bodies. Int. J. Differ. Equ. 2013, 10 (2013). Article ID. 681575CrossRefGoogle Scholar
  4. 4.
    Geiser, J., Lüskow, K.F., Schneider, R.: Levitron: multiscale analysis of stability. Dyn. Syst. 29(2), 208–224 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration illustrated by the StörmerVerlet method. Acta Numerica 12, 399–450 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics, 2nd edn. Springer, Heidelberg (2006)Google Scholar
  7. 7.
    Jiang, Y.-L., Wing, O.: A note on convergence conditions of waveform relaxation algorithms for nonlinear differential–algebraic equations. Appl. Numer. Math. 36(2–3), 281–297 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    McLachlan, R.I., Atela, P.: The accuracy of symplectic integrators. Nonlinearity 5, 541–562 (1992)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Jürgen Geiser
    • 1
    Email author
  • Karl Felix Lüskow
    • 1
  • Ralf Schneider
    • 1
  1. 1.Institute of PhysicsErnst Moritz Arndt University of GreifswaldGreifswaldGermany

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