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Error analysis of variational integrators of unconstrained Lagrangian systems

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Abstract

An error analysis of variational integrators is obtained, by blowing up the discrete variational principles, all of which have a singularity at zero time-step. Divisions by the time step lead to an order that is one less than observed in simulations, a deficit that is repaired with the help of a new past–future symmetry.

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References

  1. Bobenko, A.I., Schröder, P., Sullivan, J.M., Ziegler, G.M. (eds.): Discrete Differential Geometry. Birkhäuser Verlag, Boston (2008)

    Google Scholar 

  2. Bobenko, A.I., Suris, Y.B.: Discrete differential geometry. Consistency as integrability. arXiv:math.DG/0504358v1, (2005)

  3. Bou-Rabee N., Marsden J.E.: Hamilton–Pontryagin integrators on Lie groups part I: Introduction and structure-preserving properties. Found. Comput. Math. 9, 197–219 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cuell C., Patrick G.W.: Skew critical problems. Regul. Chaotic Dyn. 12, 589–601 (2007)

    Article  MathSciNet  Google Scholar 

  5. Cuell, C., Patrick, G.W.: Geometric discrete analogues of tangent bundles and constrained Lagrangian systems. J. Geom. Phys. 59, 976–997 (2009)

    Article  MATH  Google Scholar 

  6. Gelfand, I.M., Fomin, S.V.: Calculus of Variations. Prentice-Hall Inc., New Jersey (1963). [Revised English edition translated and edited by Richard A. Silverman]

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Patrick, G.W.: Two axially symmetric coupled rigid bodies: relative equilibria, stability, bifurcations, and a momentum preserving symplectic integrator. PhD thesis, University of California at Berkeley (1991)

  10. Patrick G.W.: Lagrangian mechanics without ordinary differential equations. Rep. Math. Phys. 57, 437–443 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Weinstein A.: Lagrangian mechanics and groupoids. Fields Inst. Comm. 7, 207–231 (1996)

    Google Scholar 

  12. Wendlandt J.M., Marsden J.E.: Discrete integrators derived from a discrete variational principle. Phys. D 106, 233–246 (1997)

    Article  MathSciNet  Google Scholar 

  13. Yoshimura H., Marsden J.E.: Dirac structures in Lagrangian mechanics. I. Implicit Lagrangian systems. J. Geom. Phys. 57, 133–156 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Yoshimura H., Marsden J.E.: Dirac structures in Lagrangian mechanics. II. Variational structures. J. Geom. Phys. 57, 209–250 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Applied Mathematics and Mathematical Physics, University of Saskatchewan, Saskatoon, Saskatchewan, S7N 5E6, Canada

    George W. Patrick & Charles Cuell

Authors
  1. George W. Patrick
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  2. Charles Cuell
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Correspondence to George W. Patrick.

Additional information

G. W. Patrick is funded by the Natural Sciences and Engineering Reseach Council, Canada.

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Patrick, G.W., Cuell, C. Error analysis of variational integrators of unconstrained Lagrangian systems. Numer. Math. 113, 243–264 (2009). https://doi.org/10.1007/s00211-009-0245-3

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  • DOI: https://doi.org/10.1007/s00211-009-0245-3

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