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Abstract : 
The aim of this research is to study the efficiency of symplectic and nonsymplectic RungeKutta methods in solving Kepler problem. The numerical behavior of the RungeKutta (RK) methods that are symmetric such as the implicit midpoint rule (IMR), implicit trapezoidal rule (ITR), 2stage and 2stage Gauss (G2) method are compared with the nonsymmetric RungeKutta methods such as the explicit and implicit Euler (EE and IE), explicit midpoint rule (EIMR), explicit trapezoidal rules (EITR), explicit 4stage RungeKutta (RK4) method and 2stage Radau IIA method (R2A). Kepler problem is one type of nonlinear Hamiltonian problem that describes the motion in a plane of a material point that is attracted towards the origin with a force inversely proportional to the distance squared. The exact solutions phase diagram produces a unit circle. The nonsymplectic methods only reproduce a unit circle at certain time intervals while the symplectic methods do produce a unit circle at any time intervals. Some phase diagram show spiral in or spiral out patterns which means the solutions are running away from the unit circle. This also means that the absolute error will be increasing in long time integration. The numerical experiments for the Kepler problem are given for many time intervals and the results show that the most efficient method is G2 of order4 and surprisingly RK4 seems to be efficient too although it is not a symplectic nor a symmetric method. The numerical results on Kepler problem concluded that, the higher the order of the method, the most efficient the method can be in solving Kepler problem despite whether they are explicit or implicit or symmetric and symplectic. 
References 
1. AfzaliFar, B., Lidström, P. A JointSpace Parametric Formulation for the Vibrations of Symmetric GoughStewart Platforms (2015) Advances in Intelligent Systems and Computing, 1089, pp. 323329. Cited 5 times. http://www.springer.com/series/11156 ISBN: 9783319084213 doi: 10.1007/9783319084220_48 2. Butcher, J.C. Numerical Methods for Ordinary Differential Equations (2016) Numerical Methods for Ordinary Differential Equations, pp. 1513. Cited 1455 times. http://onlinelibrary.wiley.com/book/10.1002/9781119121534 ISBN: 9781119121534; 9781119121503 doi: 10.1002/9781119121534 3. Butcher, J.C. Coefficients for the study of RungeKutta integration processes (Open Access) (1963) Journal of the Australian Mathematical Society, 3 (2), pp. 185201. Cited 193 times. doi: 10.1017/S1446788700027932 4. Fernández De Bustos, I., Agirrebeitia, J., Ajuria, G., Ansola, R. An alternative fullpivoting algorithm for the factorization of indefinite symmetric matrices (Open Access) (2015) Journal of Computational and Applied Mathematics, 274, pp. 4457. Cited 4 times. doi: 10.1016/j.cam.2014.07.003 5. Chan, R.P.K. On symmetric RungeKutta methods of high order (1990) Computing, 45 (4), pp. 301309. Cited 27 times. doi: 10.1007/BF02238798 6. Cong, N.h. Parallel iteration of symmetric RungeKutta methods for nonstiff initialvalue problems (Open Access) (1994) Journal of Computational and Applied Mathematics, 51 (1), pp. 117125. Cited 17 times. doi: 10.1016/03770427(94)900949 7. Calvo, M., Franco, J.M., Montijano, J.I., Rández, L. Sixthorder symmetric and symplectic exponentially fitted RungeKutta methods of the Gauss type (Open Access) (2009) Journal of Computational and Applied Mathematics, 223 (1), pp. 387398. Cited 33 times. doi: 10.1016/j.cam.2008.01.026 8. Hamilton, S.W.R. (1833) On A General Method of Expressing the Paths of Light, & of the Planets, by the Coefficients of A Characteristic Function, 1, pp. 795826. Cited 32 times. Dublin University Review and Quarterly Magazine 9. Kang, F. (1985) Canonical Difference Schemes for Hamiltonian Canonical Differential Equations, pp. 5971. Cited 2 times. International workshoon Applied Differential Equation, World Scientific Publishing 10. Ruth, R.D. A canonical integration technique (1983) IEEE Transactions on Nuclear Science, 30 (4), pp. 26692671. Cited 486 times. doi: 10.1109/TNS.1983.4332919 11. SanzSerna, J.M. Rungekutta schemes for Hamiltonian systems (1988) BIT, 28 (4), pp. 877883. Cited 250 times. doi: 10.1007/BF01954907 12. SanzSerna, J.M. Rungekutta schemes for Hamiltonian systems (1988) BIT, 28 (4), pp. 877883. Cited 250 times. doi: 10.1007/BF01954907 13. SanzSerna, J.M. Rungekutta schemes for Hamiltonian systems (1988) BIT, 28 (4), pp. 877883. Cited 250 times. doi: 10.1007/BF01954907 14. Suris, Y.B. On the conservation of the symplectic structure in the numerical solution of Hamiltonian systems. In: Numerical solution of differential equations (1988) Keldysh Inst. of Applied Mathematics, pp. 148160. Cited 35 times. Ed. S. Filippov, Moscow 15. Tang, W., Lang, G., Luo, X. Construction of symplectic (partitioned) RungeKutta methods with continuous stage (2016) Applied Mathematics and Computation, 286, pp. 279287. Cited 7 times. doi: 10.1016/j.amc.2016.04.026 16. Zhu, B., Hu, Z., Tang, Y., Zhang, R. Symmetric and symplectic methods for gyrocenter dynamics in timeindependent magnetic fields (2016) International Journal of Modeling, Simulation, and Scientific Computing, 7 (2), art. no. 1650008. Cited 2 times. http://www.worldscinet.com/ijmssc/ doi: 10.1142/S1793962316500082 
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