Testing of Adaptive Symplectic Conservative Numerical Methods for Solving the Kepler Problem
- 37 Downloads
The properties of a family of new adaptive symplectic conservative numerical methods for solving the Kepler problem are examined. It is shown that the methods preserve all first integrals of the problem and the orbit of motion to high accuracy in real arithmetic. The time dependences of the phase variables have the second, fourth, or sixth order of accuracy. The order depends on the chosen values of the free parameters of the family. The step size in the methods is calculated automatically depending on the properties of the solution. The methods are effective as applied to the computation of elongated orbits with an eccentricity close to unity.
Keywords:Hamiltonian systems symplecticity invertibility integrals of motion Runge–Kutta methods Kepler problem.
This work was supported in part by Moscow State University and the Scientific Research Institute for System Analysis of the Russian Academy of Sciences, project no. 0065-2014-0031.
- 1.G. N. Duboshin, Celestial Mechanics: Basic Problems and Methods (Nauka, Moscow, 1968; Defense Tech. Inf. Center, Fort Belvoir, 1969).Google Scholar
- 2.V. I. Arnold, Mathematical Methods of Classical Mechanics (Nauka, Moscow, 1979; Springer Science & Business Media, New York, 2013).Google Scholar
- 3.L. D. Landau and E. M. Lifshitz, Mechanics (Nauka, Moscow, 1973; Butterworth-Heinemann, Oxford, 1976).Google Scholar
- 5.Computational Molecular Dynamics: Challenges, Methods, Ideas, Ed. by P. Deuflhard, J. Hermans, B. Leimkuhler, (Springer-Verlag, Berlin, 1998).Google Scholar
- 7.Y. B. Suris, “On the conservation of the symplectic structure in numerical solutions of Hamilton systems,” Numerical Solutions of Ordinary Differential Equations (Keldysh Inst. Appl. Math., USSR Acad. Sci., Moscow, 1988), pp. 148–160.Google Scholar
- 22.W. Oewel and M. Sofrouniou, “Symplectic Runge–Kutta schemes II: Classification of symmetric methods,” Preprint (University of Paderborn, 1997).Google Scholar