本文针对不等距网格,从Rayleigh商(Rayleigh quotient)角度出发,构造了若干求解ODE特征值问题的高阶格式,并进行误差分析.文中高阶格式的构造是基于线性有限元及其对应的差分格式进行的 .单纯的线性有限元及其对应的差分格式求解PDE特征值问题都只有二阶精度,我们利用质量集中和加权组合的思想通过将二者结合得到四阶精度的算法.本文从 理论和实验的角度构造高阶格式并进行了相应的误差分析.通过在五种网格上计算四阶精度格式的误差阶系数,将四阶格式加权组合的新格式甚至可以达到六阶精度 .最后用数值实验验证了构造的高阶格式的误差阶同时,本文构造的两种四阶格式相对于传统的线性有限元方法,在同等量级误差的要求下, 需要的网格数有量级的减少.From the perspective of Rayleigh quotient, we constructed several high-order accuracy schemes to solve the ODE eigenproblem on non-uniform grid, and analyze the error. Purely linear finite element scheme and finite difference scheme to solve corresponding ODE eigenproblem are only second order accuracy, Use a weighted combination of the two and error compensation or mass lumping method to construct the fourth-order accuracy schemes. We construct high-order accuracy schemes and analyse the corresponding error both in theoretical and experimental levels. Through numerical experiments on five kinds of grid, we construct new schemes even up to sixth-order accuracy, by computing the coefficient of the specific error-order of the fourth-order accuracy schemes. At last, we use numerical experiment to verify the accuracy order of these new schemes. At the request of the same order of magnitude error, the number of-grid that our two fourth-order schemes need have a magnitude of decrease compared to the traditional linear finite element method
