常微分方程欧拉折线法的Picard迭代改进

Improvement of Euler Method for Ordinary Differential Equations by Picard Iteration

考虑常微分方程数值解法,对欧拉折线法加以改进。主要方法是,对每个节点处局部线性近似解进行一次Picard迭代修正,用局部修正解估算下一节点处解值。证明了算法局部截断误差比欧拉折线法高一阶,分析了算法的收敛性与稳定性。通过方程实例数值计算进一步验证了本文算法所得近似解比欧拉折线法近似解更接近精确解。

In this paper we study numerical algorithms for ordinary differential equations, and improve the Euler method. We refine local linear approximate solution at each node by Picard iteration, and then estimate the numerical solution at next node by the refined local approximate solution. The local truncation error of our algorithm is smaller than that of Euler method by one order. The con-vergence and stability are proven. Finally, we show that our algorithm gives approximate solution closer to the exact solution than Euler method by an example.