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Construction and analysis of higher order Galerkin variational integrators

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An Erratum to this article was published on 12 July 2016

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.

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References

  1. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, Berlin Heidelberg New York (1999)

    Google Scholar 

  2. Bou-Rabee, N., Owhadi, H.: Stochastic variational integrators. IMA J. Numer. Anal. 29, 421–443 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  4. Cortés, J., Martínez, S.: Non-holonomic integrators. Nonlinearity 14(5), 1365–1392 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. Hall, J., Leok, M.: Spectral variational integrators. (preprint, arXiv:1211.4534) (2012)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  11. Lasagni, F.M.: Canonical Runge-Kutta methods. Zeitschrift für Angewandte Mathematik und Physik ZAMP 39, 952–953 (1988). doi:10.1007/BF00945133

    Article  MathSciNet  MATH  Google Scholar 

  12. Leimkuhler, B., Reich, S.: Simulating Hamiltonian dynamics. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  13. Leok, M., Shingel, T.: General Techniques for Constructing Variational Integrators. Frontiers of Mathematics in China arXiv:1102.2685(2011)

  14. Lew, A., Marsden, J.E., Ortiz, M., West, M.: Asynchronous variational integrators. Arch. Ration. Mech. Anal. 167, 85–146 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  16. Lew, A., Marsden, J.E., Ortiz, M., West, M.: Variational time integrators. Int. J. Numer. Methods Eng. 60(1), 153–212 (2004b)

    Article  MathSciNet  MATH  Google Scholar 

  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. Leyendecker, S., Marsden, J.E., Ortiz, M.: Variational integrators for constrained dynamical systems. J. Appl. Math. Mech. 88 (9), 677–708 (2008)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  20. Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numerica 10, 357–514 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Marsden, J.E., Patrick, G.W., Shkoller, S.: Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Sci. 199, 351–395 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. McLachlan, R.I., Quispel, G.R.W.: Splitting methods. Acta Numerica 11, 341–434 (2002). doi:10.1017/S0962492902000053

    Article  MathSciNet  MATH  Google Scholar 

  23. Ober-Blöbaum, S.: Galerkin variational integrators and modified symplectic Runge-Kutta methods. submitted (2014)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  27. Reich, S.: Momentum conserving symplectic integrations. Phys. D 76(4), 375–383 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  28. Saake, N.: Konstruktion und Analyse variationeller Integratoren höherer Ordnung, diploma thesis, Paderborn (2012)

  29. Sanz-Serna, J.M.: Runge-Kutta schemes for Hamiltonian systems. BIT Numer. Math. 28, 877–883 (1988). doi:10.1007/BF01954907

    Article  MathSciNet  MATH  Google Scholar 

  30. Stern, A., Grinspun, E.: Implicit-explicit variational integration of highly oscillatory problems. SIAM Multiscale Model. Simul. 7, 1779–1794 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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

    MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  33. Suris, Y.B.: Hamiltonian methods of Runge-Kutta type and their variational interpretation. Math. Model. 2, 78–87 (1990)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  35. Yoshida, H.: Construction of higher order symplectic integrators. Phys. Lett. A 150, 262–268 (1990). doi:10.1016/0375-9601(90)90092-3

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Computational Dynamics and Optimal Control, Department of Mathematics, University of Paderborn, Warburger Str. 100, 33098, Paderborn, Germany

    Sina Ober-Blöbaum & Nils Saake

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  1. Sina Ober-Blöbaum
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  2. Nils Saake
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Correspondence to Sina Ober-Blöbaum.

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Communicated by: Axel Voigt

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Ober-Blöbaum, S., Saake, N. Construction and analysis of higher order Galerkin variational integrators. Adv Comput Math 41, 955–986 (2015). https://doi.org/10.1007/s10444-014-9394-8

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