中山大学学报自然科学版 ›› 2009, Vol. 48 ›› Issue (5): 1-5.

• 研究论文 •    下一篇

一种改进的精细-龙格库塔法

富明慧;梁华力   

  1. (中山大学应用力学与工程系, 广东 广州 510275)
  • 收稿日期:2008-10-14 修回日期:1900-01-01 出版日期:2009-09-25 发布日期:2009-09-25
  • 通讯作者: 富明慧

<An Improved Precise Runge-Kutta Integration

FU Minghui;LIANG Huali   

  1. (Department of Applied Mechanics and Engineering, Sun Yat-sen University, Guangzhou 510275,China)
  • Received:2008-10-14 Revised:1900-01-01 Online:2009-09-25 Published:2009-09-25

摘要: 提出了求解非线性动力学方程的一种改进的精细龙格库塔法。首先对于线性问题,利用等步长的Newton-Cotes积分公式计算非齐次方程Duhamel积分形式的特解。由于在此过程中提出了一种简便的算法,与常规的同精度数值积分法相比,能较大程度地降低计算量和存储量。然后将上述方法推广到非线性问题,对于各积分点上未知的状态参量,参照龙格库塔法的几何解释进行一次预估。与已有的精细龙格库塔法相比,在精度和效率上均有较大程度的改善。算例结果充分证明了该方法的有效性。

关键词: 非线性动力学方程, 精细积分法, 龙格库塔法, 指数矩阵运算, 数值积分

Abstract: An improved precise Runge-Kutta integration is presented for solving nonlinear dynamical system. First of all, Newton-Cotes integral formula is used to calculate the special solution of nonhomogeneous equation in Duhamel integral form for linear system. Since a simple and convenient calculational method is advanced, the handling technique can reduce the whole gross of computation and storage comparing to common numerical integration methods with the same precision. Then the technique is extended to nonlinear system and unknown state parameters are forecasted consulting to the geometrical explanation of Runge-Kutta method. The method has better precision and efficiency to a relatively high extent than the existed precision Runge-Kutta method and the result of the numerical examples substantiates the availability adequately.

Key words: nonlinear dynamical equation, precise time-step integration method, Runge-Kutta integration, calculation technique of matrix exponential, numerical integration

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