×

Spurious behavior of a symplectic integrator. (English) Zbl 1129.65093

Summary: We study the existence and behavior of spurious solutions of symplectic Euler method for some Hamiltonian systems. It is shown that the symplectic integrator applied to Hamiltonian systems, in general, doesn’t avoid spurious behavior, even spurious period-two solutions. The numerical results are presented.

MSC:

65P10 Numerical methods for Hamiltonian systems including symplectic integrators
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Guo, B.; Sleeman, B. D.; Chen, S., On the discrete logistic model of biology, Applicable Analysis, 33, 215-231 (1989) · Zbl 0657.92001
[2] Humphries, A. R.; Stuart, A. M., Runge-Kutta methods for dissipative and gradient dynamical systems, SIAM J. Numerical Anal., 31, 5, 1452-1485 (1994) · Zbl 0807.34057
[3] Humphries, A. R., Spurious solutions of numerical methods for initial value problems, IMA J. Numer. Anal., 13, 263-290 (1993) · Zbl 0769.65041
[4] Iserles, A.; Peplow, A. T.; Stuart, A. M., A unified approach to spurious solutions introduced by time discretisation, Part I: Basic theory, SIAM J. Numer. Anal., 28, 6, 1723-1751 (1991) · Zbl 0736.65050
[5] Iserles, A.; Stuart, A. M., Unified approach to spurious solutions introduced by time discretisation, Part II: BDF-like methods, IMA J. Numer. Anal., 12, 487-502 (1992) · Zbl 0760.65071
[6] May, R. M., Simple mathematical models with very complicate dynamics, Nature, 261, 459-467 (1976) · Zbl 1369.37088
[7] Sleeman, B. D.; Griffiths, D. F.; Mitchell, A. R.; Smith, P. D., Stable periodic solutions in nonlinear difference equations, SIAM J. Sci. Sta. Comput., 9, 543-557 (1988) · Zbl 0646.65064
[8] Stuart, A. M.; Humphries, A. R., Dynamical Systems and Numerical Analysis, (Cambridge Monographs on Applied and Computational Mathematics (1996), Cambridge University Press: Cambridge University Press Singapore) · Zbl 0869.65043
[9] Stuart, A. M.; Peplow, A. T., The dynamics of the Theta method, SIAM J. Sci. Stat. Comput., 12, 6, 1351-1372 (1991) · Zbl 0737.65061
[10] Hairer, E.; Lubich, C.; Wanner, G., Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations (2002), Springer-Verlag: Springer-Verlag Cambridge · Zbl 0994.65135
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.