Construction of seminumerical schemes: application to the artificial satellite problem
by
Roberto Barrio
Dpto. Matematica Aplicada, Univ. Zaragoza, E-50009 Zaragoza, Spain

In this communication we present the construction of seminumerical -or semianalytical- schemes in the numerical integration of systems of differential equations. The approach that we follows employs the modified perturbation method proposed by Barrio and Palacian, which uses the Lie series formalism in a way that permits to split the differential system in two parts: one that follows a Hamiltonian structure and the other one that are non-Hamiltonian. By means of these perturbative scheme we apply the averaging techniques in such a way that for the Hamiltonian part the transformation is a symplectic one. Therefore, for the combination of the averaging process and a symplectic numerical integrator, the total integration process will preserve the Hamiltonian structure. As, in general, the equations suitable for the averaging process (periodic standard form) are not written as separable Hamiltonians, the symplectic integrators for these equations are implicit. Thus, we also compare with other integrators such as symmetric integrators. In particular we use collocation Runge-Kutta methods based on Chebyshev polynomials, that permits a dense output in the form of a Chebyshev series, situation useful in some applications. A remarkable numerical property of these methods is the A-stability and P-stability.

All the techniques are applied to the important problem of the orbit analysis of Earth artificial satellites subject to Hamiltonian (Earth potential) and non-Hamiltonian perturbations (the air drag). We have computed a second order average system by means of the perturbative scheme for low eccentric orbits and we give the numerical integration result as a Chebyshev series of low degree by means of the Chebyshev RK method. Besides, the algebraic structure of the kernel of the averaging process is established.

Date received: January 28, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeb-32.