ABSTRACT

A class of modified collocation Runge–Kutta–Nyström (RKN) methods for solving second-order initial value problem $y^{″}=f(t,y,y^{\prime })$, $y(t_0)=y_0$, $y^{\prime }(t_0)=y_0^{\prime }$ has been formulated and studied in the paper. The new methods are applicable to a larger class of second-order initial value problems compared to the classical collocation RKN methods and they reduce to the classical methods when $y^{\prime }$ is absent from the equation. Superconvergence for the new methods is attained when the set of collocation points satisfies orthogonality conditions. We proved that an s-stage modified collocation RKN method is of accuracy order of at least s for any set of collocation parameters $\mathit{c}$ and at most $2s$ when $\mathit{c}$ are Gauss points. The stability function and the stability of some modified RKN methods have also been investigated. Numerical experiments are included to demonstrate the advantage of the new methods.

摘要

一类求解二阶初值问题的改进搭配Runge-Kutta-Nyström（RKN）方法${\mathrm{是的}}^{”}=F(吨,\mathrm{是的},{\mathrm{是的}}^{\text{'}})$,$\mathrm{是的}({吨}_0)={\mathrm{是的}}_0$,${\mathrm{是的}}^{\text{'}}({吨}_0)={\mathrm{是的}}_0^{\text{'}}$已在论文中进行了阐述和研究。与经典搭配 RKN 方法相比，新方法适用于更大类的二阶初值问题，并且当${\mathrm{是的}}^{\text{'}}$等式中不存在。当配置点集合满足正交性条件时，新方法的超收敛性得到了实现。我们证明了一个s阶段改进的搭配 RKN 方法对于任何搭配参数集的精度至少为s$\mathit{C}$并且最多$2s$什么时候$\mathit{C}$是高斯点。还研究了一些改进的 RKN 方法的稳定性函数和稳定性。包括数值实验以证明新方法的优势。