Evaluation of Numerical Methods for Aircraft Trajectory Computation

Abstract

Aircraft trajectories are used by ground automation systems that support Air Traffic Management (ATM) operations. The performance of these operations depends on the accuracy of the trajectories. This paper presents an analysis of numerical methods used by trajectory predictors to solve the equations of motion. The accuracy of numerical methods for Ordinary Differential Equations (ODEs) depend on the order of the approximations when evaluating the derivatives to be integrated. There is a tradeoff between accuracy and computational efficiency. It is found that standard methods, such as 2nd order Runge-Kutta, are adequate for nominal conditions. However, discontinuities in some of the terms in the equations for vertical rate and the presence of vertical wind gradients may benefit from more sophisticated methods. A new method, the Adaptive Altitude (AdAL) step method, that changes the independent variable from time to altitude is proposed and evaluated. The AdAL method yields a performance comparable to 2nd order, but at half the computational load. Results are presented for climb and descent scenarios under different wind conditions. Accuracy and numerical stability metrics are presented. Operational impact of errors is discussed.