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BIT Numerical Mathematics

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24-11-2017

Lagrangian and Hamiltonian Taylor variational integrators

Authors: Jeremy Schmitt, Tatiana Shingel, Melvin Leok

Published in: BIT Numerical Mathematics | Issue 2/2018

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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.

previous article Ubiquitous evaluation of layer potentials using Quadrature by Kernel-Independent Expansion

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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)

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)MathSciNetMATH

Ford, J.: The Fermi–Pasta–Ulam problem: paradox turns discovery. Phys. Rev. 213(5), 271–310 (1992)MathSciNet

Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration, Volume 31 of Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (2006)MATH

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

Iserles, A.: A First Course in the Numerical Analysis of Differential Equations. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2009)

Iserles, A., Norsett, S.P.: On quadrature methods for highly oscillatory integrals and their implementation. BIT Numer. Math. 44(4), 755–772 (2004)MathSciNetCrossRefMATH

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)MathSciNetCrossRefMATH

Keller, H.B.: Numerical Methods for Two-Point Boundary Value Problems. Dover Publications Inc., New York (1992)

10.

Lall, S., West, M.: Discrete variational Hamiltonian mechanics. J. Phys. A 39(19), 5509–5519 (2006)MathSciNetCrossRefMATH

11.

Leok, M., Shingel, T.: Prolongation–collocation variational integrators. IMA J. Numer. Anal. 32(3), 1194–1216 (2012)MathSciNetCrossRefMATH

12.

Leok, M., Shingel, T.: General techniques for constructing variational integrators. Front. Math. China 7(2), 273–303 (2012)MathSciNetCrossRefMATH

13.

Leok, M., Zhang, J.: Discrete Hamiltonian variational integrators. IMA J. Numer. Anal. 31(4), 1497–1532 (2011)MathSciNetCrossRefMATH

14.

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

15.

Neidinger, R.D.: Introduction to automatic differentiation and MATLAB object-oriented programming. SIAM Rev. 52(3), 545–563 (2010)MathSciNetCrossRefMATH

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)MathSciNetCrossRefMATH

17.

Rall, L.B.: Automatic Differentiation: Techniques and Applications. Lecture Notes in Computer Science, vol. 120. Springer, Berlin (1981)CrossRef

18.

Revels, J., Lubin, M., Papamarkou, T.: Forward-mode automatic differentiation in Julia. In: AD2016—7th International Conference on Algorithmic Differentiation (2016)

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)

22.

Sommeijer, B.P.: An explicit Runge–Kutta method of order twenty-five. CWI Q. 11(1), 75–82 (1998)MathSciNetMATH

23.

Stern, A., Grinspun, E.: Implicit-explicit variational integration of highly oscillatory problems. Multiscale Model. Simul. 7(4), 1779–1794 (2009)MathSciNetCrossRefMATH

24.

Tan, X.: Almost symplectic Runge–Kutta schemes for Hamiltonian systems. J. Comput. Phys. 203, 250–273 (2005)MathSciNetCrossRefMATH

Title: Lagrangian and Hamiltonian Taylor variational integrators
Authors: Jeremy Schmitt
Tatiana Shingel
Melvin Leok
Publication date: 24-11-2017
Publisher: Springer Netherlands
Published in: BIT Numerical Mathematics / Issue 2/2018
Print ISSN: 0006-3835
Electronic ISSN: 1572-9125
DOI: https://doi.org/10.1007/s10543-017-0690-9

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