Convergence Analysis of the Modified Chebyshev-Picard Iteration Algorithm
Abstract
The topic of this thesis is the analysis of the ability of Modified Chebyshev-Picard Iteration (MCPI) to converge on an accurate solution. The convergence analysis includes a discussion of MCPI and a brief explanation of the derivation of first and second order MCPI. The limitations and benefits of MCPI are also discussed. MCPI has historically used relative error to evaluate if the algorithm has converged to a solution. This thesis presents a means of evaluating absolute error of the solution. The Absolute Error Analysis is derived for first and second order MCPI and the difference between it and relative error is discussed using an example problem. The different means of evaluating convergence are then used to evaluate how changes to MCPI affect convergence of a nonlinear oscillator. The convergence is evaluated by inspecting the error profiles of the converged solution, the number of iterations necessary to converge to the solution, and the maximum final time which converges to a solution. MCPI convergence is then analyzed when using two-body motion for three predefined orbital trajectories. The convergence is analyzed according to the error profiles of each trajectory, the number of iterations necessary to converge for each trajectory, and the maximum final time at which each trajectory can converge to a solution. The use of first and second order MCPI is discussed along with the use of Cartesian Coordinates and Orbital Elements and their affect on the convergence of MCPI. The three trajectories are then analyzed with the introduction of J2 and drag perturbations, both individually and combined effects. The perturbed trajectories are evaluated using Multi-segment MCPI to extend the convergence window of MCPI. The variation in MCPI variables used to compare with the number of iterations necessary to converge and the error profiles of each segment to evaluate the efficacy of Multi-segment MCPI as a numerical integration tool. The entirety of the MCPI code usage was developed for universal implementation and is available to the public.
Citation
Moore, Mason (2021). Convergence Analysis of the Modified Chebyshev-Picard Iteration Algorithm. Master's thesis, Texas A&M University. Available electronically from https : / /hdl .handle .net /1969 .1 /195158.