微分求积法的特性及其改进

摘要

微分求积法已在科学和工程计算中得到了广泛应用。然而，有关时域微分求积法的数值稳定性、计算精度即阶数等基本特性，仍缺乏系统性的分析结论。依据微分求积法的基本原理，推导证明了微分求积法的权系数矩阵满足 V-变换这一重要特性；利用微分求积法和隐式 Runge-Kutta 法的等值性，证明了时域微分求积法是A-稳定、s 级 s 阶的数值方法。在此基础上，为进一步提高传统微分求积法的计算精度，利用待定系数法和 Padé逼近，推导出了一类新的 s 级2s 阶的微分求积法。数值计算对比结果验证了所提出的新微分求积法比传统的微分求积法具有更高的计算精度。%The differential quadrature method has been widely used in scientific and engineering computa-tion.However,for the basic characteristics of time domain differential quadrature method,such as numerical stability and calculation accuracy or order,are still lack of systematic analysis conclusions. According to the principle of differential quadrature method,it has been derived and proved that the weight coefficient matrix of the differential quadrature method meets the important V-transformation feature.Through the equivalence of differential quadrature method and implicit Runge-Kutta method,it has been proved that the differential quadrature method is A-stable and s-stage s-order method.On this basis,in order to further improve the accuracy of the time domain differential quadrature method,a class of improved differential quadrature method of s-stage 2s-order has been derived using undetermined coefficients method and Padéapproximations.The numerical results show that the proposed differential quadrature method is more precise than the traditional differential quadrature method.