Duffing方程的辛精细积分方法研究

SYMPLECTIC PRECISE INTEGRATION METHOD FOR DUFFING EQUATION

辛精细积分方法汲取了辛几何算法保持动力学系统辛结构的优点和精细积分方法高精度的数值优点,其实现过程中涉及到大量矩阵求逆运算.为减小辛精细积分方法的运算量,本文在辛精细积分算法之前先将非齐次方程近似齐次化,使得矩阵求逆部分不显含时间,降低矩阵求逆计算量,并将这一方法应用于无阻尼Duffing方程的数值分析.通过与经典四阶Runge-Kutta格式及精细积分方法对比,发现辛精细积分方法在数值精度、计算耗时、保持系统能量等方面明显优于Runge-Kutta格式.此外,与精细积分方法相比,辛精细积分方法在保持系统能量方面存在明显优势.

The symplectic precise integration method owns the advantages of the symplectic method and the precise integration method. In the implementation procedure of which, matrix inversion is a time-consuming step. Aiming at this problem, we homogenize the inhomogeneous equation approximately before the design of the symplectic precise integration method in this paper. The homogenizing process makes the matrix inversion time-independent and reduces the calculated quantity of the matrix inversion process, which is used in the symplectic precise integration method of the non-damping Duffing equation in this paper. From the numerical results, we can conclude that： The symplectic precise integration method is superior to the classic Runge-Kutta method in the nu- merical precisi energy can on, the energy-preserving property and time-consuming; Comparing with the symplectic method, be preserved in the numerical simulation of the symplectic precise integration method.