# Lagrangian and Hamiltonian Taylor variational integrators

- 147 Downloads

## Abstract

In this paper, we present a variational integrator that is based on an approximation of the Euler–Lagrange boundary-value problem via Taylor’s method. This can be viewed as a special case of the shooting-based variational integrator. The Taylor variational integrator exploits the structure of the Taylor method, which results in a shooting method that is one order higher compared to other shooting methods based on a one-step method of the same order. In addition, this method can generate quadrature nodal evaluations at the cost of a polynomial evaluation, which may increase its efficiency relative to other shooting-based variational integrators. A symmetric version of the method is proposed, and numerical experiments are conducted to exhibit the efficacy and efficiency of the method.

## Keywords

Geometric numerical integration Variational integrators Symplectic integrators Hamiltonian mechanics## Mathematics Subject Classification

37M15 65P10 70H05## References

- 1.Bucker, H.M., Lang, B., Mey, D. Bischof, C.: Bringing together automatic differentiation and OpenMP. In: ICS ’01 Proceedings of the 15th International conference on Supercomputing, pp. 246–251 (2001)Google Scholar
- 2.Butcher, J.C.: On fifth and sixth order Runge–Kutta methods: order conditions and order barriers. Can. Appl. Math. Q.
**17**(3), 433–445 (2009)MathSciNetzbMATHGoogle Scholar - 3.Ford, J.: The Fermi–Pasta–Ulam problem: paradox turns discovery. Phys. Rev.
**213**(5), 271–310 (1992)MathSciNetGoogle Scholar - 4.Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration, Volume 31 of Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
- 5.Haro, A.: Automatic differentiation methods in computational dynamical systems. In: New Directions Short Course: Invariant Objects in Dynamical Systems and their Application, Minneapolis, June 2011. Institute for Mathematics and its Applications (IMA). http://www.maia.ub.edu/~alex/ima/ima1.pdf
- 6.Iserles, A.: A First Course in the Numerical Analysis of Differential Equations. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2009)Google Scholar
- 7.Iserles, A., Norsett, S.P.: On quadrature methods for highly oscillatory integrals and their implementation. BIT Numer. Math.
**44**(4), 755–772 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Jorba, A., Zou, M.: A software package for the numerical integration of ODEs by means of high-order Taylor methods. Exp. Math.
**14**, 99–117 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Keller, H.B.: Numerical Methods for Two-Point Boundary Value Problems. Dover Publications Inc., New York (1992)Google Scholar
- 10.Lall, S., West, M.: Discrete variational Hamiltonian mechanics. J. Phys. A
**39**(19), 5509–5519 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Leok, M., Shingel, T.: Prolongation–collocation variational integrators. IMA J. Numer. Anal.
**32**(3), 1194–1216 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Leok, M., Shingel, T.: General techniques for constructing variational integrators. Front. Math. China
**7**(2), 273–303 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Leok, M., Zhang, J.: Discrete Hamiltonian variational integrators. IMA J. Numer. Anal.
**31**(4), 1497–1532 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer.
**10**, 357–514 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Neidinger, R.D.: Introduction to automatic differentiation and MATLAB object-oriented programming. SIAM Rev.
**52**(3), 545–563 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Patterson, M., Weinstein, M., Rao, A.: An efficient overloaded method for computing derivatives of mathematical functions in MATLAB. ACM Trans. Math. Softw.
**39**(3), 17 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Rall, L.B.: Automatic Differentiation: Techniques and Applications. Lecture Notes in Computer Science, vol. 120. Springer, Berlin (1981)CrossRefGoogle Scholar
- 18.Revels, J., Lubin, M., Papamarkou, T.: Forward-mode automatic differentiation in Julia. In: AD2016—7th International Conference on Algorithmic Differentiation (2016)Google Scholar
- 19.Schmitt, J.M., Leok, M.: Properties of Hamiltonian variational integrators. IMA J. Numer. Anal.
**36**, drx010 (2017). https://doi.org/10.1093/imanum/drx010 - 20.Shen, X., Leok, M.: Geometric exponential integrators. arXiv:1703.00929 [math.NA] (2017)
- 21.Simo, C.: Principles of Taylor methods for analytic, non-stiff, ODE. In: Advanced Course on Long Time Integration. University of Barcelona (2007)Google Scholar
- 22.Sommeijer, B.P.: An explicit Runge–Kutta method of order twenty-five. CWI Q.
**11**(1), 75–82 (1998)MathSciNetzbMATHGoogle Scholar - 23.Stern, A., Grinspun, E.: Implicit-explicit variational integration of highly oscillatory problems. Multiscale Model. Simul.
**7**(4), 1779–1794 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Tan, X.: Almost symplectic Runge–Kutta schemes for Hamiltonian systems. J. Comput. Phys.
**203**, 250–273 (2005)MathSciNetCrossRefzbMATHGoogle Scholar