Picard Iteration method, Chebyshev Polynomial Approximation, and Global Numerical Integration of Dynamical Motions

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Abstract
The Picard iteration method and the Chebyshev polynomial approximation were combined to obtain numerically a global solution of ordinary differential equations. The method solves both the initial and boundary value problems. The method directly provides not the tabulated values of the solution but the polynomials interpolating the solution in the integration interval given. In the case of scalar computation, the method is a few to several times as fast as the multistep method when (1) a good approximation of the solution is known beforehand, (2) the right hand members of differential equations are weakly dependent on the solution, and/or (3) the magnitude of the right hand members are small. Thus the method is suitable for (1) orbital improvements, (2) the integration of almost uniform rotations, and (3) perturbed dynamics in general. The method will be greatly accelerated by using vector/parallel computers since its main part is the numerical quadrature of known function of time. (C) 1997 American Astronomical Society.

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