A Robust Implicit Optimal Order Formula for Direct Integration of Second Order Orbital Problems

Abstract

In this paper, a robust implicit formula of optimal order for direct integration of general second order orbital problems of ordinary differential equations (ODEs) is proposed. This method is considered capable avoiding the computational burden and wastage in computer time in connection with the method of reduction to first order systems. The integration algorithms and analysis of the basic properties are based on the adoption of Taylor’s expansion and Dahlquist stability model test. The resultant integration formula is of order ten and it is zero-stable, consistent, convergent and symmetric. The numerical implementation of the method to orbital and two-body problems demonstrates increased accuracy with the same computational effort on comparison with similar second order formulas.

Keywords: Optimal-order, Zero-stability, Convergence, Consistent, IVPs, Predictor-corrector, Error constant, Symmetric.