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Advances in Computational Mathematics

, Volume 41, Issue 6, pp 955–986 | Cite as

Construction and analysis of higher order Galerkin variational integrators

  • Sina Ober-BlöbaumEmail author
  • Nils Saake
Article

Abstract

In this work we derive and analyze variational integrators of higher order for the structure-preserving simulation of mechanical systems. The construction is based on a space of polynomials together with Gauss and Lobatto quadrature rules to approximate the relevant integrals in the variational principle. The use of higher order schemes increases the accuracy of the discrete solution and thereby decrease the computational cost while the preservation properties of the scheme are still guaranteed. The order of convergence of the resulting variational integrators is investigated numerically and it is discussed which combination of space of polynomials and quadrature rules provide optimal convergence rates. For particular integrators the order can be increased compared to the Galerkin variational integrators previously introduced in Marsden and West (Acta Numerica 10:357–514 2001). Furthermore, linear stability properties, time reversibility, structure-preserving properties as well as efficiency for the constructed variational integrators are investigated and demonstrated by numerical examples.

Keywords

Discrete variational mechanics Numerical convergence analysis Symplectic methods Variational integrators 

Mathematics Subject Classifications (2010)

37J99 65Lxx 70H25 

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References

  1. 1.
    Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, Berlin Heidelberg New York (1999)Google Scholar
  2. 2.
    Bou-Rabee, N., Owhadi, H.: Stochastic variational integrators. IMA J. Numer. Anal. 29, 421–443 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Campos, C.M., Junge, O., Ober-Blöbaum, S.: Higher order variational time discretization of optimal control problems In: 20th International Symposium on Mathematical Theory of Networks and Systems, Melbourne (2012)Google Scholar
  4. 4.
    Cortés, J., Martínez, S.: Non-holonomic integrators. Nonlinearity 14(5), 1365–1392 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fetecau, R.C., Marsden, J.E., Ortiz, M., West, M.: Nonsmooth Lagrangian Mechanics and Variational Collision Integrators. SIAM J. Appl. Dyn. Syst. 2(3), 381–416 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hairer, E., Wanner, G.: Solving ordinary differential equations II: Stiff and differential-algebraic problems. Springer Series in Computational Mathematics. Springer, Heidelberg (2010). http://opac.inria.fr/record=b1130632 Google Scholar
  7. 7.
    Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. Springer Series in Computational Mathematics, vol. 31. Springer, Berlin Heidelberg New York (2002)Google Scholar
  8. 8.
    Hall, J., Leok, M.: Spectral variational integrators. (preprint, arXiv:1211.4534) (2012)
  9. 9.
    Kane, C., Marsden, J.E., Ortiz, M., West, M.: Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. Int. J. Numer. Methods Eng. 49(10), 1295–1325 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kobilarov, M., Marsden, J.E., Sukhatme, G.S.: Geometric discretization of nonholonomic systems with symmetries. Discret Contin Dyn. Syst. - Series S 1(1), 61–84 (2010)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Lasagni, F.M.: Canonical Runge-Kutta methods. Zeitschrift für Angewandte Mathematik und Physik ZAMP 39, 952–953 (1988). doi: 10.1007/BF00945133 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Leimkuhler, B., Reich, S.: Simulating Hamiltonian dynamics. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  13. 13.
    Leok, M., Shingel, T.: General Techniques for Constructing Variational Integrators. Frontiers of Mathematics in China arXiv:1102.2685(2011)
  14. 14.
    Lew, A., Marsden, J.E., Ortiz, M., West, M.: Asynchronous variational integrators. Arch. Ration. Mech. Anal. 167, 85–146 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lew, A., Marsden, J.E., Ortiz, M., West, M.: An overview of variational integrators. In: Franca, L.P., Tezduyar, T.E., Masud, A. (eds.) Finite element methods: 1970’s and beyond, CIMNE, pp. 98–115 (2004a)Google Scholar
  16. 16.
    Lew, A., Marsden, J.E., Ortiz, M., West, M.: Variational time integrators. Int. J. Numer. Methods Eng. 60(1), 153–212 (2004b)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Leyendecker, S., Ober-Blöbaum, S.: A variational approach to multirate integration for constrained systems. In: Samin, J.C., Fisette, P. (eds.) Multibody dynamics, computational methods in applied sciences, vol. 28, pp. 97–121. Springer Netherlands (2013), doi: 10.1007/978-94-007-5404-1_5
  18. 18.
    Leyendecker, S., Marsden, J.E., Ortiz, M.: Variational integrators for constrained dynamical systems. J. Appl. Math. Mech. 88 (9), 677–708 (2008)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Leyendecker, S., Ober-Blöbaum, S., Marsden, J.E., Ortiz, M.: Discrete mechanics and optimal control for constrained systems. Optim. Control, Appl. Methods 31(6), 505–528 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numerica 10, 357–514 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Marsden, J.E., Patrick, G.W., Shkoller, S.: Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Sci. 199, 351–395 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    McLachlan, R.I., Quispel, G.R.W.: Splitting methods. Acta Numerica 11, 341–434 (2002). doi: 10.1017/S0962492902000053 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ober-Blöbaum, S.: Galerkin variational integrators and modified symplectic Runge-Kutta methods. submitted (2014)Google Scholar
  24. 24.
    Ober-Blöbaum, S., Junge, O., Marsden, J.E.: Discrete mechanics and optimal control: an analysis. Control. Optimisation Calc. Var. 17(2), 322–352 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ober-Blöbaum, S., Tao, M., Cheng, M., Owhadi, H., Marsden, J.E.: Variational integrators for electric circuits. J. Comput. Phys. 242, 498–530 (2013). doi: 10.1016/j.jcp.2013.02.006 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Patrick, G.W., Cuell, C.: Error analysis of variational integrators of unconstrained Lagrangian systems. Numer. Math. 113, 243–264 (2009). doi: 10.1007/S00211-009-0245-3 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Reich, S.: Momentum conserving symplectic integrations. Phys. D 76(4), 375–383 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Saake, N.: Konstruktion und Analyse variationeller Integratoren höherer Ordnung, diploma thesis, Paderborn (2012)Google Scholar
  29. 29.
    Sanz-Serna, J.M.: Runge-Kutta schemes for Hamiltonian systems. BIT Numer. Math. 28, 877–883 (1988). doi: 10.1007/BF01954907 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Stern, A., Grinspun, E.: Implicit-explicit variational integration of highly oscillatory problems. SIAM Multiscale Model. Simul. 7, 1779–1794 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Sun, G.: Symplectic partitioned Runge-Kutta methods. J. Comput. Math. 11(4), 365–372 (1993) http://www.jcm.ac.cn/EN/abstract/article_8879.shtml MathSciNetzbMATHGoogle Scholar
  32. 32.
    Suris, Y.B.: The canonicity of mappings generated by Runge-Kutta type methods when integrating the systems \(\ddot {x}=-\partial u/\partial x\). USSR Comput. Math. Math. Phys. 29(1), 138–144 (1989) doi: 10.1016/0041-5553(89)90058-X MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Suris, Y.B.: Hamiltonian methods of Runge-Kutta type and their variational interpretation. Math. Model. 2, 78–87 (1990)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Tao, M., Owhadi, H., Marsden, J.E.: Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging. Multiscale Model. Simul. 8(4), 1269–1324 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Yoshida, H.: Construction of higher order symplectic integrators. Phys. Lett. A 150, 262–268 (1990). doi: 10.1016/0375-9601(90)90092-3 MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Computational Dynamics and Optimal Control, Department of MathematicsUniversity of PaderbornPaderbornGermany

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