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Generating Function Methods for Coefficient-Varying Generalized Hamiltonian Systems

Published online by Cambridge University Press:  03 June 2015

Xueyang Li*
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University, Xiangtan 411105, Hunan, China
Aiguo Xiao
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University, Xiangtan 411105, Hunan, China
Dongling Wang
Affiliation:
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University, Xiangtan 411105, Hunan, China
*
*Corresponding author. Email: lixy1217@163.com
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Abstract

The generating function methods have been applied successfully to generalized Hamiltonian systems with constant or invertible Poisson-structure matrices. In this paper, we extend these results and present the generating function methods preserving the Poisson structures for generalized Hamiltonian systems with general variable Poisson-structure matrices. In particular, some obtained Poisson schemes are applied efficiently to some dynamical systems which can be written into generalized Hamiltonian systems (such as generalized Lotka-Volterra systems, Robbins equations and so on).

Type
Research Article
Copyright
Copyright © Global-Science Press 2014

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References

[1]Austin, M., Krishnaprasad, P. S. and Wang, L. S., Almost poisson integration of rigid body systems, J. Comput. Phys., 107 (1993), pp. 105117.CrossRefGoogle Scholar
[2]Chen, C. M., Tang, Q. and Hu, S. F., Finite element method with superconvergence for nonlinear Hamiltonian systems, J. Comput. Math., 29 (2011), pp. 167184.Google Scholar
[3]Cohen, D. and Hairer, E., Linear energy-preserving integrators for Poisson systems, BIT Numer. Math., 51 (2011), pp. 91101.CrossRefGoogle Scholar
[4]Ergenc, T. and Karasozen, B., Poisson integrators for Volterra lattice equations, Appl. Numer. Math., 56 (2005), pp. 870887.Google Scholar
[5]Feng, K. and Qin, M. Z., Symplectic Geometric Algorithms for Hamiltonian Systems, Springer-Verlag, 2010.CrossRefGoogle Scholar
[6]Feng, K. and Wang, D. L., Symplectic difference schemes for Hamiltonian systems in general symplectic structures, J. Comput. Math., 9 (1991), pp. 8696.Google Scholar
[7]Feng, K., Wu, H. M. and Qin, M. Z., Symplectic difference schemes for the linear Hamiltonian canonical systems, J. Comput. Math., 8 (1990), pp. 371380.Google Scholar
[8]Feng, K., Wu, H. M., Qin, M. Z. and Wang, D. L., Construction of canonical difference schemes for Hamiltonian formalism via generating functions, J. Comput. Math., 7 (1989), pp. 7196.Google Scholar
[9]Hairer, E., Lubich, C. and Wanner, G., Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer-Verlag, 2002.Google Scholar
[10]HernáNdez-Bermejo, B. and FairéN, V., Hamiltonian structure and Darboux Theorem for families of generalized Lotka-Volterra systems, J. Math. Phys., 39 (1998), pp. 61626174.CrossRefGoogle Scholar
[11]Hong, J. L. and Sun, Y. J., Generating functions of multi-symplectic RK methods via DW Hamilton-Jacobi equations, Numer. Math., 110 (2008), pp. 491519.CrossRefGoogle Scholar
[12]Jay, L. O., Preserving Poisson structure and orthogonality in numerical integration of differential equations, Computers Math. Appl., 48 (2004), pp. 237255.CrossRefGoogle Scholar
[13]Li, J. B., Zhao, X. H. and Liu, Z. R., Theory and Application of Generalized Hamiltonian Systems, Science Press, 1994.Google Scholar
[14]Liu, X. S., Qi, Y. Y., He, J. F. and Ding, P. Z., Recent progress in symplectic algorithms for use in quantum systems, Commun. Comput. Phys., 2 (2007), pp. 153.Google Scholar
[15]Makazaga, J. and Murua, A., A new class ofsymplectic integration schemes based on generating functions, Numer. Math., 113 (2009), pp. 631642.CrossRefGoogle Scholar
[16]Marsden, J. E. AND Ratiu, S. T., Introduction to Mechanics and Symmetry, SpringerVerlag, 1994.CrossRefGoogle Scholar
[17]Olver, P. J., Applications of Lie Groups to Differential Equations, Springer-Verlag, 1993.CrossRefGoogle Scholar
[18]Tang, T. and Xu, X., Accuracy enhancement using spectral postprocessing for differential equations and integral equations, Commun. Comput. Phys., 5 (2009), pp. 779792.Google Scholar
[19]Wang, D. L., Poisson difference schemes for Hamiltonian systems on Poisson manifolds, J. Comput. Math., 9 (1991), pp. 115124.Google Scholar
[20]Zhang, S. Y. AND Deng, Z. C., Geometric Integration Theory and Application of Nonlinear Dynamical Systems, Northwestern Polytechnical University Press, 2005.Google Scholar

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