Abstract

This chapter provides a simple Adaptive Local Variational Iteration Method (ALVIM) that can efficiently solve nonlinear differential equations and orbital problems of spacecrafts. Based on a general first-order form of nonlinear differential equations, the iteration formula is analytically derived and then discretized using Chebyshev polynomials as basis functions in the time domain. It leads to an iterative numerical algorithm that only involves the addition and multiplication of sparse matrices. Moreover, the Jacobian matrix is free from inversing. Apart from that, a straightforward adaptive scheme is proposed to refine the configuration of the algorithm, involving the length of time steps and the number of collocation nodes in a time step. With the adaptive scheme, the prescribed accuracy can be guaranteed without manually tuning the configuration of the algorithm. Since the refinement is adjusted automatically, our algorithm reduces overcalculation for smooth and slowly changing problems. Examples such as large amplitude pendulum and perturbed two-body problem are used to verify this easy-to-use adaptive method's high accuracy and efficiency.