×

Hamiltonian algorithms for Hamiltonian systems and a comparative numerical study. (English) Zbl 0900.65212


MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
Full Text: DOI

References:

[1] Feng, K., On difference schemes and symplectic geometry, (Feng, K., Proc. 1984 Beijing Symposium on Differential Geometry and Differential Equations — Computation of Partial Differential Equations (1985), Science Press: Science Press Beijing), 42 · Zbl 0659.65118
[2] Feng, K., Canonical difference schemes for Hamiltonian canonical differential equations, (Proc. Int. Workshop on Applied Differential Equations. Proc. Int. Workshop on Applied Differential Equations, June 1985, Beijing (1986), World Scientific: World Scientific Singapore), 59 · Zbl 0626.70018
[3] Feng, K., Difference schemes for Hamiltonians formalism and symplectic geometry, J. Comput. Math., 4, 279 (1986) · Zbl 0596.65090
[4] Feng, K., Symplectic geometry and numerical methods in fluid dynamics, (Zhuang, F. G.; Zhu, Y. L., Proc. 10th Int. Conf. Numerical Methods in Fluid Dynamics. Proc. 10th Int. Conf. Numerical Methods in Fluid Dynamics, Beijing, 1986. Proc. 10th Int. Conf. Numerical Methods in Fluid Dynamics. Proc. 10th Int. Conf. Numerical Methods in Fluid Dynamics, Beijing, 1986, Lecture Notes in Physics, Vol. 264 (1987), Springer: Springer Berlin), 1 · Zbl 0626.76002
[5] Feng, K.; Qin, M., The symplectic methods for the computation of Hamiltonian equations, (Zhu, Y.; Gu, B.-y., Numerical Methods for Partial Differential Equations. Numerical Methods for Partial Differential Equations, Lecture Notes in Mathematics No. 1297 (1987), Springer: Springer Berlin), 1, Shanghai, 1986 · Zbl 0639.70007
[6] Feng, K.; Wu, H.-m.; Qin, M.-z.; Wang, D.-l., Construction of canonical difference schemes for Hamiltonian formalism via generating functions, J. Comput. Math., 7, 71 (1989) · Zbl 0681.70020
[7] Feng, K., The Hamiltonian way for computing Hamiltonian dynamics, (Spigler, R., Proc. State of the Art of Applied and Industrial Mathematics. Proc. State of the Art of Applied and Industrial Mathematics, Venice, 1989 (1990), Kluwer: Kluwer Dordrecht) · Zbl 0718.58026
[8] Feng, K.; Wu, H.-m.; Qin, M.-z., Symplectic difference schemes for the linear Hamiltonian canonical systems, J. Comput. Math., 8, 371 (1990) · Zbl 0712.70032
[9] Ge, Z.; Feng, K., On the approximation of linear Hamiltonian systems, J. Comput. Math., 6, 88 (1988) · Zbl 0656.70016
[10] Li, C.-w.; Qin, M.-z., A symplectic difference scheme for the infinite dimensional Hamiltonian system, J. Comput. Math., 6, 164 (1988) · Zbl 0669.70019
[11] Qin, M.-z., A symplectic difference scheme for the Hamiltonian equation, J. Comput. Math., 5, 203 (1987) · Zbl 0633.65133
[12] M.-z. Qin, D.-l. Wang and M.-q. Zhang, Explicit symplectic difference schemes for separable Hamiltonian systems, J. Comput. Math., to appear.; M.-z. Qin, D.-l. Wang and M.-q. Zhang, Explicit symplectic difference schemes for separable Hamiltonian systems, J. Comput. Math., to appear. · Zbl 0745.65050
[13] Wu, Y.-h., Symplectic transformation and symplectic difference schemes, Chin. J. Numer. Math. Appl., 12, 23 (1990)
[14] Wu, Y.-h., The generating function for the solution of ODE and its discrete methods, Comput. Math. Appl., 15, 1041 (1988) · Zbl 0654.65050
[15] M.-q. Zhang and M.-z. Qin, A note on convergence of symplectic schemes, J. Comput. Math., to appear.; M.-q. Zhang and M.-z. Qin, A note on convergence of symplectic schemes, J. Comput. Math., to appear.
[16] Ge, Z.; Marsden, J. E., Lie-Poisson Hamiltonian-Jacobi theory and Lie-Poisson integrator, Phys. Lett. A, 133, 134 (1988) · Zbl 1369.70038
[17] Qin, M.-z., Canonical difference scheme for the Hamiltonian equation, Math. Methods Appl. Sci., 11, 543 (1989) · Zbl 0683.70018
[18] Qin, M.-z.; Zhang, M.-q., Multi-stage symplectic schemes of two kinds of Hamiltonian system for wave equation, Comput. Math. Appl., 19, 10, 51 (1990) · Zbl 0695.65072
[19] M.-z. Qin and M.-q. Zhang, Symplectic Runge-Kutta schemes for Hamiltonian systems, J. Comput. Math., to appear.; M.-z. Qin and M.-q. Zhang, Symplectic Runge-Kutta schemes for Hamiltonian systems, J. Comput. Math., to appear. · Zbl 1069.65140
[20] Qin, M.-z.; Zhang, M.-q., Explicit Runge-Kutta-like schemes to solve certain quantum operator equations of motion, J. Stat. Phys., 60, 839 (1990) · Zbl 1086.81538
[21] K. Feng and D.-l. Wang, A note on conservation laws of symplectic difference schemes for Hamiltonian systems, J. Comput. Math., to appear.; K. Feng and D.-l. Wang, A note on conservation laws of symplectic difference schemes for Hamiltonian systems, J. Comput. Math., to appear. · Zbl 0745.65047
[22] D.L. Wang, Poisson difference schemes for Hamiltonian systems on Poisson manifolds, J. Comput. Math., to appear.; D.L. Wang, Poisson difference schemes for Hamiltonian systems on Poisson manifolds, J. Comput. Math., to appear. · Zbl 0738.65061
[23] Arnold, V. I., Mathematical methods of Classical Mechanics (1974), Nauka: Nauka Moscow · Zbl 0647.70001
[24] Feynman, R., (Lectures on Physics, vol. 1 (1963), Addison-Wesley: Addison-Wesley Reading, MA), Ch. 9 · Zbl 0138.43404
[25] Buneman, O., Ideal gas dynamics in Hamiltonian form with benefit for numerical schemes, Phys. Fluids, 23, 1716 (1980) · Zbl 0439.76058
[26] Ruth, R. D., A canonical integration technique, IEEE Trans. Nucl. Sci., 30, 2669 (1983)
[27] Sanz-Serna, J. M., Runge-Kutta schemes for Hamiltonian systems, BIT, 28, 877 (1988) · Zbl 0655.70013
[28] Sanz-Serna, J. M., The numerical integration of Hamiltonian systems, (Cash, J. R., Proc. Conf. on Computational Differential equations. Proc. Conf. on Computational Differential equations, Imperial College, London (1989), Oxford Univ. Press: Oxford Univ. Press Oxford), to appear · Zbl 0766.65058
[29] Suris, Y. B., On the preservation of the symplectic structure for numerical integration of Hamiltonian systems, (Fillipov, S. S., Numerical Solution of Ordinary Differential Equations (1988), Keldysh Institute of Applied Mathematics, USSR Academy of Science: Keldysh Institute of Applied Mathematics, USSR Academy of Science Moscov), 148, (in Russian) · Zbl 0786.34021
[30] Lasagni, F. M., Canonical Runge-Kutta methods, Z. Angew. Math. Phys., 39, 952 (1988) · Zbl 0675.34010
[31] P.J. Channel and C. Scovel, Symplectic integration of Hamiltonian systems, Los Alamos National Laboratory Preprint La-UR-88-1828.; P.J. Channel and C. Scovel, Symplectic integration of Hamiltonian systems, Los Alamos National Laboratory Preprint La-UR-88-1828.
[32] Chierchia, L.; Gallavotti, G., Smooth prime integrals for quasi-integrable Hamiltonian systems, Nuovo Cimento B, 67, 277 (1982)
[33] Pöschel, J., Integribility of Hamiltonian systems on cantor sets, Commun. Pure Appl. Math., 35, 653 (1982) · Zbl 0542.58015
[34] Chernikov, A. A.; Sagdeev, R. Z.; Zaslavsky, G. M., Stochastic webs, Physica D, 33, 65 (1988) · Zbl 0696.58034
[35] Arnold, V. I., Remarks on quasicrystallic symmetries, Physica D, 33, 21 (1988) · Zbl 0684.52008
[36] Beloshapkin, V. V.; Chernikov, A. A.; Natenzon, M. Y.; Petrovichev, B. A.; Sagdeev, R. Z.; Zaslavsky, G. M., Chaotic streamlines in preturbulent states, Nature, 337, 543 (1989)
[37] Tang, Y.-f.; Feng, K., Symplectic computation of geodesic flows on closed surfaces and Kepler motion (1990), Academia Sinica Computing Center, preprint
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.