Chebyshev collocation methods for fast orbit determination
type: Item
instance of: scholarly article
title: Chebyshev collocation methods for fast orbit determination
zbMATH Open document ID: 0944.65086
DOI: 10.1016/S0096-3003(98)00003-4
author: Roberto Barrio
author: Manuel Palacios
author: Antonio Elipe
published in: Applied Mathematics and Computation
publication date: 2000-09-13T00:00:00Z
review text: The goal of this paper is an application of the Chebyshev collocation method to numerical integration of the initial value problem for nonhomogeneous Kepler ordinary differential equation. The nonhomogenity term depends on the solution \(r\) and its time derivative denoted here by \(v\), but it is assumed to be a small perturbation of the homogeneous Kepler equation. The original equation is transformed to the so-called Dziobek-Brouwer form, which is more convenient for the applied numerical method. This numerical problem is related to determination of the orbit of the artificial Earth satellite; hence the long physical time computation should be done in relatively short computing time. In such situation a stable and fast solver is required. Following the ideas of \textit{C. W. Clenshaw} [Proc. Cambridge Philos. Soc. 53, 134-149 (1957; Zbl 0077.32503); J. Soc. Invest. Appl. Math., Ser. B, Numer. Anal. 1, 26-37 (1964; Zbl 0136.36303)], the authors propose to apply the method based on the Fourier-Chebyshev approximation of the right-hand-side of the equation. More exactly, the integral equation of the form \[ y(t)=y(t_k)+\int_{t_k}^tf(s,y(s))ds \] is replaced by the implicit finite difference scheme for the approximate value \(\widetilde y(t)\) of \(y(t)\) \[ \widetilde y(t)=y_k+\sum_{j=0}^n a_jT_j(x). \] Here \(x\in [-1,1]\) corresponds to the variable \(t\) and coefficients \(a_j\) depend on the Fourier coefficients of the function \(f\) computed approximately by means of the Gauss-Lobatto quadrature formulas, where knots are taken at extremal points of the Chebyshev polynomial of the first kind, \(T_{n-1}\). This method enables to compute densely the approximate solution. Since the scheme is implicit, iterations are necessary; the authors use the simple Picard method. The integration step can be taken very large, namely equal to the approximate period of the solution. Neither stability nor convergence theorems are given, however numerical experiments show the effic…
reviewed by: Krzysztof Moszyński
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 65L12
Mathematics Subject Classification ID: 65L60
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 70M20
zbMATH DE Number: 1311108
zbMATH Keywords: Fourier-Chebyshev approximation
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Chebyshev collocation method
zbMATH Keywords: Kepler ordinary differential equation
zbMATH Keywords: Earth satellite
zbMATH Keywords: implicit finite difference scheme
zbMATH Keywords: Picard method
zbMATH Keywords: numerical experiments
MaRDI profile type: Publication
cites work: Q4376870
cites work: Perturbations in rectangular coordinates by iteration
cites work: Q5619659
cites work: Q5849046
cites work: Q3242438
cites work: The solution of nonlinear ordinary differential equations in Chebyshev series
cites work: A family of embedded Runge-Kutta formulae
cites work: Q5720147
cites work: Q5533821
cites work: Solving Ordinary Differential Equations I
cites work: A family of implicit Chebyshev methods for the numerical integration of second-order differential equations
cites work: Some remarks on multivariate Chebyshev polynomials
cites work: Q3868817
description: scientific article; zbMATH DE number 1311108