An Adaptive Local Variational Iteration Method for Orbit Propagation in Astrodynamics Problems

In this paper, a highly accurate and efficient Adaptive Local Variational Iteration Method (ALVIM) is presented to fulfil the need of the astrodynamics society for fast and accurate computational methods for guidance and control. The analytical iteration formula of this method is derived by using a general form of the first order nonlinear differential equations, followed by straightforward discretization using Chebyshev polynomials and collocation. The resulting numerical algorithm is very concise and easy to use, only involving highly sparse matrix operations of addition and multiplication, and no inversion of the Jacobian is required. Apart from the simple yet efficient iteration formula, a straightforward adaptive scheme is introduced to refine the step size and the collocation nodes at each time segment. The presented adaptive method guarantees prescribed accuracy without manual tuning of the algorithm. The computational cost of ALVIM, in terms of functional evaluations, is 1–2 orders of magnitude lower than adaptive finite difference methods. Numerical results of a large amplitude pendulum, perturbed two-body problem, and three-body problem validate the high accuracy and efficiency of this easy-to-use adaptive method.