二阶常微分方程边值问题的5点差分算法

Five-Point Difference Algorithm for Second-Order Ordinary Differential Boundary Value Problems

摘要: 本文针对二阶线性常微分方程两点边值问题，通过具有最高代数精度的二阶导数的组合式近似，构造了具有最高阶精度的等步长的五点差分格式，并利用广义Peano定理给出了精度阶数。用构造的格式对多个算例编程计算，用Taylor展开方法对边界进行离散处理，并将计算结果与精确解比较。计算结果表明，所构造的差分格式达到了最高的八阶精度。

Abstract: In this paper, for the two-point boundary value problems of second-order linear ordinary differential equations, a five-point difference scheme with equal step is constructed by constructing a combined approximation of the second derivative with the highest algebraic precision, and the precision order is given by using the generalized Peano theorem. A lot of examples are calculated with program by the constructed scheme, applying respectively the Taylor expansion method to discrete the boundary. The numerical results are compared with the exact solutions and show that the constructed scheme achieves the highest order accuracy of eight.