非线性动力方程的一种改进精细积分单步方法

An improved precise integration single-step method for nonlinear dynamic equations

微分求积法和单步块方法都是单步多级数值方法,但是直接应用于求解非线性动力方程时的计算量比较巨大,为此提出了一种基于单步块方法的改进精细积分单步方法。结合精细积分法,该方法采用s级的单步块方法的第s个方程对Duhamel积分项进行数值积分。具体采用四阶Runge-Kutta法获得待求变量的预估值,并采用新四点积分公式计算Duhamel积分项。相对于现有的单步方法,该改进算法在数值精度和稳定性上更优。通过非线性动力方程的典型算例验证了该算法的优势。

Differential quadrature method and single-step block method are single-step multistage numerical methods, but the amount of calculation is huger when they are directly applied to solve nonlinear dynamic equations. Here, an improved precise integration single-step method based on the single-step block method was proposed. Combined with the precise integration method, this method adopted the s;equation of the s-level single-step block method to numerically integrate Duhamel integral term. Specifically, the fourth-order Runge-Kutta method was used to obtain the predicted value of the variable to be solved, and the new 4-point integral formula was used to calculate Duhamel integral term. It was shown that compared with the existing single-step method, the proposed improved algorithm has better numerical accuracy and stability;its advantages are verified with typical examples of nonlinear dynamic equations.