Fast and accurate adaptive collocation iteration method for orbit dynamic problems

Abstract

For over half a century, numerical integration methods based on finite difference, such as the Runge-Kutta method and the Euler method, have been popular and widely used for solving orbit dynamic problems. In general, a small integration step size is always required to suppress the increase of the accumulated computation error, which leads to a relatively slow computation speed. Recently, a collocation iteration method, approximating the solutions of orbit dynamic problems iteratively, has been developed. This method achieves high computation accuracy with extremely large step size. Although efficient, the collocation iteration method suffers from two limitations: (A) the computational error limit of the approximate solution is not clear; (B) extensive trials and errors are always required in tuning parameters. To overcome these problems, the influence mechanism of how the dynamic problems and parameters affect the error limit of the collocation iteration method is explored. On this basis, a parameter adjustment method known as the “polishing method” is proposed to improve the computation speed. The method proposed is demonstrated in three typical orbit dynamic problems in aerospace engineering: a low Earth orbit propagation problem, a Molniya orbit propagation problem, and a geostationary orbit propagation problem. Numerical simulations show that the proposed polishing method is faster and more accurate than the finite-difference-based method and the most advanced collocation iteration method.