Chebyshev collocation methods for fast orbit determination
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 efficiency of this approach. The paper contains results of computations for \(n=22,32,40,70\).
scientific article; zbMATH DE number 1311108