基于泰勒展开的线性多步法构造方法的研究

A Study on Construction for Linear Multi-Step Methods Based on Taylor Expansion

摘要: 本文在前人对常微分方程初值问题的线性多步法公式研究的基础上，对于线性多步法公式中基于泰勒展开的构造方法进行了探究。我们尝试使用加权平均法构造得出了一个新的公式，随后对此给出了实例进行检验，并对构造生成新公式的原先两个公式和新公式进行了对照分析。经数值实验，该方法具有可行性，某些所得新公式在求解常微分方程初值问题中具有较高稳定性和较小误差。

Abstract: Following the previous literature on multi-step formulae for initial value problems of ODEs (ordi-nary differential equations), we study the construction for linear multi-step methods based on Taylor expansion in this paper. We try the weighted average method and derive a new formula. Then we check this new method in an example, and compare the original two formulae and con-structed new formula. Via such numerical experiment, this method is reliable. Some constructed new formulae can have relatively high stability and small error while solving ODEs initial value problems.