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Abstract: In this paper, precise integration is compounded with Runge–Kutta method and a new effective integration method is presented for solving nonlinear dynamical system. Arbitrary dynamical system can be expressed as nonhomogeneous linear differential equation problem. Precise integration is used to solve its corresponding homogeneous equation and Runge–Kutta methods can be used to calculate the nonhomogeneous, nonlinear items. The precise integration may have large time step and the time step of the Runge–Kutta methods can be adjusted to improve the computational precision. The handling technique in this paper not only avoids the matrix inversion but also improves the stability of the numerical method. Finally, the numerical examples are given to demonstrate the validity and effectiveness of the proposed method. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]