Geometric integration methods for general nonlinear dynamic equation based on Magnus and Fer expansions

Based on Magnus or Fer expansion for solving linear differential equation and operator semi-group theory, Lie group integration methods for general nonlinear dynamic equation are studied. Approximate schemes of Magnus type of 4th, 6th and 8th order are constructed which involve only 1,4 and 10 different commutators, and the time-symmetry properties of the schemes are proved. In the meantime, the integration methods based on Fer expansion are presented. Then by connecting the Fer expansion methods with Magnus expansion methods some techniques are given to simplify the construction of Fer expansion methods. Furthermore time-symmetric integrators of Fer type are constructed. These methods belong to the category of geometric integration methods and can preserve many qualitative properties of the original dynamic system.