﻿ 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

Vol.04 No.04(2015), Article ID:16357,7 pages
10.12677/AAM.2015.44042

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

Zhiyuan Huang1, Zhijun Hu2, Cheng Wang3

1Guangzhou University, Guangzhou Guangdong

2Guangxi Normal University, Guilin Guangxi

3Huizhou University, Huizhou Guangdong

Received: Oct. 30th, 2015; accepted: Nov. 13th, 2015; published: Nov. 18th, 2015

ABSTRACT

Following the previous literature on multi-step formulae for initial value problems of ODEs (ordinary 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 constructed 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.

Keywords:Initial Value Problems, Linear Multi-Step Methods, Stability, Weighted Average Method

1广州大学，广东 广州

2广西师范大学，广西 桂林

3惠州学院，广东 惠州

1. 引言

2. 用加权平均方法构造新的线性多步法公式

2.1. 线性多步法的局部截断误差

(2.1.1)

，由Taylor展开，有

(2.1.2)

(2.1.3)

(2.1.4)

(2.1.5)

2.2. 用加权平均方法构造新公式的基本思想

(2.2.1)

(2.2.2)

, , , ,.

(2.2.3)

(2.2.4)

(2.2.5)

(2.2.6)

(2.2.7)

(2.2.8)

2.3. 数值实验

(2.3.1)

(初值问题(2.3.1)有精确解。)

Table 1. Image result data of, , and

Figure 1. Curve: Image result of and

Figure 2. Curve: Image result of and

Figure 3. Curve: Image result of and

3. 结论

A Study on Construction for Linear Multi-Step Methods Based on Taylor Expansion[J]. 应用数学进展, 2015, 04(04): 336-342. http://dx.doi.org/10.12677/AAM.2015.44042

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