Abstract
A modified Chebyshev Picard iteration method is proposed for solving orbit propagation initial/boundary value problems. Cosine sampling techniques, known as Chebyshev-Gauss-Lobatto (CGL) nodes, are used to reduce Runge’s phenomenon that plagues many series approximations. The key benefit of using the CGL data sampling is that the nodal points are distributed nonuniformly, with dense sampling at the beginning and ending times. This problem can be addressed by a nonlinear time transformation and/or by utilizing multiple time segments over an orbit. This paper suggests a method, called a multisegment method, to obtain accurate solutions overall regardless of initial states and albeit eccentricity by dividing the given orbit into two or more segments based on the true anomaly.
Highlights
A modified Chebyshev Picard iteration (MCPI) is an iterative numerical method for approximating solutions of linear or nonlinear ordinary differential equations to obtain time histories of system state trajectories [1, 2]
The orthogonality of the Chebyshev basis functions implies that the least-square approximations can be computed to arbitrary precision without a matrix inversion; the coefficients are conveniently and robustly computed from discrete inner products [7]
This problem is overcome by introducing a multisegment method and the results are compared with the basic MCPI algorithm
Summary
A modified Chebyshev Picard iteration (MCPI) is an iterative numerical method for approximating solutions of linear or nonlinear ordinary differential equations to obtain time histories of system state trajectories [1, 2]. Similar approximation approaches that use Legendre polynomials can be utilized, but the authors obtain slightly better results because the starting and ending points of the fits are not sampled as densely as the MCPI algorithm, and importantly the location of the nodes for the Chebyshev basis functions is computed exactly without iterations. This problem is overcome by introducing a multisegment method and the results are compared with the basic MCPI algorithm. The performance of the proposed approach is established by numerical examples of the two-body problem
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