非线性动力方程的三次样条-增维精细算法

提出一种针对非线性动力方程的改进精细积分方法。该方法是在时间步长内采用分段的三次样条函数拟合非齐次项,保持高精度拟合的同时避免了求导运算和高次多项式插值带来的Runge现象。通过引入4×2个变量将动力方程增加四维转化为齐次方程,并建立相应的通解格式,避免了状态空间下系统矩阵求逆。将指数矩阵分为四个子模块,利用各模块的特点分别进行理论推导及基于精细积分法进行分步、分块计算得到相应的理论解和高精度数值解,无需反复计算整个指数矩阵,提高了解算效率。针对含未知状态量的非齐次项,引入预测-校正的方法进行迭代求解。数值计算结果表明了本文方法的有效性。

Increment-dimensional precise integration method based on cubic spline interpolation for nonlinear dynamic equation

An improved precise integration method for nonlinear dynamic equations was proposed in this paper.The inhomogeneous terms were fitted by employing piecewise cubic spline interpolation at ever time-step to increase fitting precision and to avoid calculating differential coefficient,curve oscillation of high-order polynomial fitting was also avoided.The original nonlinear dynamic equations were converted into homogenous equations with four increased dimensions by importing 4×2 variables,the relevant calculating format was built without inversing the state-space system matrix.Based on the mathematic feature of homogenized matrix,a partitioning and step by step precise integration calculating method was presented by theoretical deducing and blocked calculation,no need for computing matrix exponential at every time step,and the efficiency was improved.Predict-correct method based on Taylor format was used for inhomogeneous terms with state variables.The numerical examples show that high precision and fast speed were achieved.