LIE-POISSON INTEGRATORS FOR A RIGID SATELLITE ON A CIRCULAR ORBIT | Atılım Üniversitesi Açık Erişim Sistemi
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LIE-POISSON INTEGRATORS FOR A RIGID SATELLITE ON A CIRCULAR ORBIT
Oluşturulma Tarihi: 16-09-2015
Niteleme Bilgileri
Yazar(lar): Aydin , Ayhan (Yazar),
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Yayın Tarihi: 16-09-2015
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Lie-Poisson integrators, symplectic integrators, rigid body equations, splittings.
Abstract. In the last two decades, many structure preserving numerical methods like Poisson integrators have been investigated in numerical studies. Since the structure matrices are different in many Poisson systems, no integrator is known yet to preserve the Poisson structure of any Poisson system. In the present paper, we propose Lie– Poisson integrators for Lie–Poisson systems whose structure matrix is different from the ones studied before. In particular, explicit Lie-Poisson integrators for the equations of rotational motion of a rigid body (the satellite) on a circular orbit around a fixed gravitational center have been constructed based on the splitting. The splitted parts have been composed by a first, a second and a third order compositions. It has been shown that the proposed schemes preserve the quadratic invariants of the equation. Numerical results reveal the preservation of the energy and agree with the theoretical treatment that the invariants lie on the sphere in long–term with different orders of accuracy
KaynakçaReferences[1] Hairer, E., Lubich, C. and Wanner, G., (1987), Geometric Numerical Integration Structure–PreservingAlgorithms for Ordinary Differential Equations, Springer (31), New York.[2] Antone, V. and Gladwell, I., (2004), Performance of Variable Step Size Methods for Solving ModelSeparable Hamiltonian Systems, Mathematical and Computer Modelling, 40, 1245–1262.[3] Tocino, A. and Vigo–Aguiar, J., (2005), Symplectic Conditions for Exponential Fitting Runge–Kutta–Nystr¨om Methods, Mathematical and Computer Modelling, 42, 873–876.[4] Reich, S., (1996), Symplectic Integrators for Systems of Rigid Bodies, In Integration algorithms andclassical mechanics, Fields Inst. Commun. 10, Amer. Math. Soc., 181–191.[5] Channell, P.J. and Scovel, J.C., (1991), Integrators for Lie–Poisson dynamical systems, Physica D,50, 80–88.[6] Ge, Z.G. and Marsden, J.E., (1998), Lie–Poisson–Hamiltonian theory and Lie–Poisson integrators.Physics Letter A. 133(3), 134–139.[7] McLachlan, R.I., (1993), Explicit Lie–Poisson Integration and the Euler Equations. Physics ReviewE. 71, 3043–3046.[8] Breiter, S., Nesvorn´y, D. and Vokrouhlick´y, D., (2005), Efficient Lie–Poisson Integrator for SecularSpin Dynamics of Rigid Bodies. The Astronomical Journal, 130, 1267–1277.[9] Austin, M.A., Krishnaprasad, P.S. and Wang, L.S., (1993), Almost Poisson Intagration of Rigid BodySystem. Journal of Computational Physics. 107, 105–117.[10] Ergen¸c, T. and Karas¨ozen, B., (2006), Poisson integrators for Volterra lattice equations. AppliedNumerical Mathematics. 56, 879–887.[11] Jay, L.O., (2004), Preserving poisson structure and orthogonality in numerical integration of differentialequations. Computers and Mathematics with Applications. 48, 237–255.[12] Aydın, A., (1998), Poisson Integrators for Completely Integrable Hamiltonian Systems. M.Sc. thesis,Department of Mathematics, Middle East Technical University, Ankara.[13] Dullin, H.R., (2004), Poisson integrator for Symmetric Rigid Bodies. Regular and Chaotic Dynamics,9(3), 255–264.[14] Karas¨ozen, B., (2004), Poisson integrators. Mathematical and Computer Modelling, 40, 1225–1244.[15] Aydın, A. and Karas¨ozen, B., (2007), Symplectic and multi-symplectic methods for coupled nonlinearSchrdinger equations with periodic solutions. Computer Physics Communications, 177, 566–583.[16] Aydın, A. and Karas¨ozen, B., (2008), Symplectic and multisymplectic Lobatto methods for the goodBoussinesq equation. Journal of Mathematical Physics, 49, 083509(1)–18.[17] Aydın, A., (2009), Multisymplectic integration of N–coupled nonlinear Schr¨odinger equation withdestabilized periodic wave solutions, Chaos, Solitons and Fractals, 41, 735751.A. AYDIN: LIE-POISSON INTEGRATORS FOR A RIGID SATELLITE ON A CIRCULAR ORBIT 161[18] Leimkuhler, B. and Reich, S., (2004), Simulating Hamitonian Dynamics, Cambridge UniversityPress:Cambridge.[19] Bhardwaj, R. and Kaur, P., (2006), Satellite’s Motion under the Effect of Magnetic Torque. AmericanJournal of Applied Sciences, 3(6), 1899–1902.[20] Maciejewski, A.J., (1995), Non–Integrability of the Planar Oscillations of a Satellite. Acta Astronomica,45, 327–344.[21] Maciejewski, A. J., (2001), Non-integrability of a certain problem of rotational motion of a rigid satellite.In Dynamics of Natural and Artificial Celestial Bodies (Edited by Pretka–Ziomek, Richardson),Kluwer Academic Publish, 187–192.[22] Maciejewski, A. J., (1995), The Observer-New method for Numerical Integration of Differential Equationsin the Presence of First Integrals. In From Newton to Chaos, (Edited by A. E. Roy and B. A.Steves), Plenum, 503–512.[23] Bogoyavlensky, O.I., (1992), Euler equations on finete–dimensional Lie coalgebras arising in problemsof mathematical physics. in Russian, Uspekhi Math. Nauka, 47(1), 107–146; English translation inRussian Math. Surveys 47(1), 117–189.[24] Feng, K., (1987), Lecture Notes in Numerical Methods for P.D.E.´s, Springer–Verlag: New York/Berlin.[25] Li, S.T. and Qin, M., (1995), Lie–Poisson Integration for rigid body dynamics. Computers and Mathematicswith Applications, 30, 105–118.[26] Feng, K., (1984), On difference schemes and symplectic geometry, Beijing Symposium On DifferentialEquations, (Edited by F. Kang): Beijing.[27] Qin, M. and Zhang, M., (1990), Multi-Stage Symplectic Schemes Of Two Kinds Of HamiltonianSystems For Wave Equations. Computers and Mathematics with Applications, 19(10), 51–62.[28] McLachlan, R.I. and Quispel, G.R.W., (2002), Splitting Methods, Acta Numerica, 71, 341–434.[29] McLachlan, R.I. and Scovel, C., (1995), Equivariant Constrained Symplectic Integration. J. NonlinearScience. 5, 233–256.[30] Reich, S., (1994), Momentum Conserving Symplectic Integrator, Physica D, 76, 375–383.[31] Cooper, GJ., (1987), Stability of Runge-Kutta methods for trajectory problems, IMA J. Numer. Anal.7, 1-13.[32] Suzuki, M. and Umeno, K., (1993), Computer Simulation Studies in Condense Matter Physics IV.,Springer, Berlin.[33] Ismail, M.S. and Alamri, S.Z., (2004), Haighly accurate finite difference method for coupled nonlinearSchr¨odinger equation. Journal of Computer and Mathematics. 81(3), 333–351.
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