Mathematical Problems in Engineering

Volume 2015, Article ID 290781, 7 pages

http://dx.doi.org/10.1155/2015/290781

## Multisegment Scheme Applications to Modified Chebyshev Picard Iteration Method for Highly Elliptical Orbits

^{1}Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, USA^{2}Royce Wisenbaker Chair in Engineering, Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, USA

Received 10 April 2014; Accepted 22 July 2014

Academic Editor: Ker-Wei Yu

Copyright © 2015 Donghoon Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

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]. In contrast to many step-by-step integrators, the MCPI algorithm approximates long arcs of the state trajectory with an iterative path approximation approach and is ideally suited to parallel computation [3]. It is well known that Picard iteration has theoretical guarantees for converging to the solution assuming the forces are continuous, once differentiable, and the solution of the differential equation is unique [4]. The rate of convergence of Picard iteration is geometric rather than quadratic for Jacobian based methods. However, given a good starting approximation, excellent efficiency is possible, and the case for parallelization provides a significant advantage [5, 6].

Orthogonal Chebyshev polynomials are used as basis functions during each path iteration, and the integrations of Picard iteration are then performed analytically. 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]. 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. The MCPI algorithm utilizes a vector-matrix framework for computational efficiency. Additionally, all Chebyshev coefficients and integrand function evaluations are independent, meaning that they can be simultaneously computed in parallel for further decreased computational costs [3].

For the MCPI algorithm, the cosine sampling techniques, known as Chebyshev-Gauss-Lobatto (CGL) nodes [8], are utilized to reduce Runge’s phenomenon. The Runge phenomenon is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree [9]. Since dense sample points are distributed at the beginning and ending locations, less accurate solutions are usually obtained where sample points are more uniformly distributed [10].

For the most extreme counterexample, let us consider an unperturbed two-body problem, where the initial position is not located near the periapsis. Obviously, large errors can be observed near the periapsis due to sparse sample points where the dynamics are most nonlinear, yet we waste the dense sample points at apoapsis when the problem is most linear as shown in Figure 1.