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Generating Function Methods for Coefficient-Varying Generalized Hamiltonian Systems

Published online by Cambridge University Press:  03 June 2015

Xueyang Li ,
Aiguo Xiao  and
Dongling Wang
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: [email protected]
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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).

Keywords

65P1065L0537M1570H05Generalized Hamiltonian systemsPoisson manifoldsgenerating functionsstructure preserving algorithmsgeneralized Lotka-Volterra systems
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 6 , Issue 1 , February 2014 , pp. 87 - 106
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. 105–117.Google 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. 167–184.Google Scholar
[3]Cohen, D. and Hairer, E., Linear energy-preserving integrators for Poisson systems, BIT Numer. Math., 51 (2011), pp. 91–101.Google Scholar
[4]Ergenc, T. and Karasozen, B., Poisson integrators for Volterra lattice equations, Appl. Numer. Math., 56 (2005), pp. 870–887.Google Scholar
[5]Feng, K. and Qin, M. Z., Symplectic Geometric Algorithms for Hamiltonian Systems, Springer-Verlag, 2010.Google Scholar
[6]Feng, K. and Wang, D. L., Symplectic difference schemes for Hamiltonian systems in general symplectic structures, J. Comput. Math., 9 (1991), pp. 86–96.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. 371–380.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. 71–96.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. 6162–6174.Google 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. 491–519.Google Scholar
[12]Jay, L. O., Preserving Poisson structure and orthogonality in numerical integration of differential equations, Computers Math. Appl., 48 (2004), pp. 237–255.Google 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. 1–53.Google Scholar
[15]Makazaga, J. and Murua, A., A new class ofsymplectic integration schemes based on generating functions, Numer. Math., 113 (2009), pp. 631–642.Google Scholar
[16]Marsden, J. E. AND Ratiu, S. T., Introduction to Mechanics and Symmetry, SpringerVerlag, 1994.Google 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. 779–792.Google Scholar
[19]Wang, D. L., Poisson difference schemes for Hamiltonian systems on Poisson manifolds, J. Comput. Math., 9 (1991), pp. 115–124.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