Recent advance in numerical integration methods of non-linear dynamics

Euler firstly established the numerical method by FDM to solve the first order nonlinear ordinary differential equations, in which the system equations only satisfied at the end of time step, and in the steps are assumed to be satisfied some pointed function. In this paper, by introducing the Jacobina matrix existed for an arbitrary autonomous non-linear dynamics, the nonlinear differential equation is reconstructed as an equivalent non-linear ordinary differential equation, in which the main terms, linear part, construct the continuous linearization equation, the remains represent high-order remainder of the nonlinear function. The proposed method doesn't changes the nature of the systems without any assumption, the nonlinear equations are reconstructed as linearized equations including the high-order remainder, Based on the CIM of an exact solution for linear systems via iteration, the equations are continuously satisfied not in discrete form, in the time steps the high accuracy solution converge quickly within given accuracy. The strategy of calculating the exponential function matrix is using summation of matrices to instead of multiplication by itself, and the matrix continuously multiple a vector. The unconditional stability has been easily proved. A lot of numerical examples verify the proposed method is simple and has generalized acceptability, high efficiency and enough accuracy.