Symplectic Effect for the Numerical Solution of Conservative Systems
Abstract
The focus of this paper is to use those numerical tools for conservative systems that provides an approximation flow for the Hamiltonian system, which defines a worldwide physical systems including planetary motion, simple Pendulum and several models. It has been observed that the symplectic scheme is found very effective for astronomical many body systems. We are particularly interested in those numerical schemes that possess the qualitative behavior of such systems and symplecticity of the flow. In this paper, we investigate the Hamiltonian systems for its symplecticity and G-symplecticity numerically and show explicitly how these techniques be effective for the preservation of energy. Since we have applied this scheme for the planetary body motion and found that the results are very much effective and the energy preserves during the motion of the planetary bodies. Since the aim of this paper is to investigate the energy preservation adopted by the symplectic methods.
References
[2] J. C. Butcher. Numerical Methods for Ordinary Differential Equations, John Wiley, ISBN 978 − 0 − 471 − 96758 − 3. Dahlquist, Germund (1963)
[3] J. C. Butcher. Implicit Runge-Kutta process.Math, Comp. 8 : 50 − 64, 1964,.
[4] J. C. Butcher. The numerical analysis of ordinary differential equations, Runge-Kutta and general linear methods, (Wiley, Chichester and New York (1987)).
[5] J. C. Butcher. Coefficients for the study of Runge-Kutta integration processes, J. Austral. Math. Soc.,3(1963)185 − 201.
[6] J. C. Butcher. General linear methods: A survey, 1985.
[7] J. C. Butcher. Yousaf Habib, Adrian Hill and Terence Norton. The control of Parasitism in G-symplectic methods, 25November2012.
[8] H.Herman, Goldstine, ”A special stability problem for linear multistep methods”, BIT 3 : 2743, doi:10.1007/BF 01963532, ISSN 0006 − 3835, 1977.
[9] Moulton, R.Forest. ” Numerical integration of ordinary differential equations”, American Mathematical Monthly (Mathematical Association of America) 33(9): 455460, doi : 10.2307/2299609, JSTOR2299609, (1926),
[10] E. Hairer, C. Lubich and G. Wanner.Geometric Numerical Intergeration Structure Pre-Serving Algorithms for Ordinary Differential Equations, Springer, secondedition, 2005.
[11] A. Iserles. A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, ISBN 978 − 0 − 521 − 55655 − 2.
[12] J. R. Dormand, and P. J. Prince. High order embedded Runge-Kutta formulae, J. Comput. Appl. Math.7(1981), no.1, 67 − 75.
[13] M. Amer Qureshi. Efficient Numerical Integeration for Gravitational N-Body Simulation. Phd Thesis, The University of Auchland, Department of Mathematics, 2010.
[14] J. M. Sanz-Serna. Symplectic integerators for Hamiltonian problems: an overveiw, Universidad de Valla.
[15] J. M. Sanz-Serna and M. P. Calvo. Numerical Hamiltonian Problems. Chapman and Hal, first edition, 1994
[16] J. M. Sanz-Serna and L. Abia. Order Conditions for Canonical Runge-Kutta schemes,
[17] SIAM, J.Num.Anal,, 28 : 1081 − 1096, 1991.
[18] Y. B. Suris. Preservation of symplectic structures in the numerical solution of Hamiltonian sytems,S.S.Fillippovesed, (Akad. Nauk, Russia). 148 − 160, 1998.
[19] Y. F. Tang. The symplecticity of multistep methods. Computers Math, Applic. 25 : 83 − 90, 1993.
[20] W. M. Wright. General linear methods with inherent Runge-Kutta stability, Phd Thesis, The University of Auchland, Department of Mathematics, 2002.
[21] Yousaf Habib. Long-term behaviour of G-symplectic methods, Phd Thesis, The University of Auchland, Department of Mathematics, 2010.
[22] Y. Habib, R. Mustafa, M. A. Qureshi. G-Symplectic Integeration of Many Body Problems,Bulletin of the Iranian Mathematical Society. (2018)44 : 937 -954.
[23] Junaid Ahmad, Yousaf Habib, Shafiq ur Rehman, Azqa Arif, Saba Shafiq and Muhammad Younas. Symplectic Effective Order Numerical Methods for Seperable Hamiltonian Systems, Symmetry. 2019, 11(2), 142
