Group preserving schemes for nonlinear dynamic system based on RKMK methods

In the present paper, the computation for the nonlinear dynamic system is discussed. Firstly the nonlinear dynamic system is converted into an augmented dynamic system in the Minkowski space, which results in the system of Lie type locally. Then Runge-Kutta Munthe-Kaas (RKMK) methods are used to the new augmented dynamic system and group-preserving integration scheme for the augmented dynamic system is constructed, in which the precise integration method is used to compute the exponential mapping. In the process of computation, the numerical schemes are formulated for nonlinear dynamic system directly. The advantages of the method presented in this paper lie in not only its group-preserving character but also in its simplicity.