﻿ 椭圆及热传导界面问题浸入界面方法的研究 Immersed Interface Method for Elliptic Interface and Heat Conduction Problem

Vol.04 No.02(2015), Article ID:15227,13 pages
10.12677/AAM.2015.42019

Immersed Interface Method for Elliptic Interface and Heat Conduction Problem

Liping Zhang, Jianping Zhao*, Shuai Zhang

College of Mathematics and System Sciences, Xinjiang University, Urumqi Xinjiang

*通讯作者。

Email: *zhaojianping@126.com

Received: Apr. 23rd, 2015; accepted: May 10th, 2015; published: May 15th, 2015

ABSTRACT

The study of interface problems has important application background. In this paper immersed interface method is improved, and especially the difficulty about the interface and the area near the interface is discussed. We use this method for solving the one-dimensional elliptic and heat equation. At last based on MATLAB, we give some numerical experiments in order to show the correctness and efficiency of the scheme. The modified immersed interface method can be used for more complicated interface problems.

Keywords:Elliptic Interface Problem, Taylor Expansion, Consistent Grid Subdivision, Heat Transfer Interface Problem

Email: *zhaojianping@126.com

1. 引言

2. 界面问题及浸入界面方法简介

2.1. 界面问题

2.2. 浸入界面方法介绍

(1)

(2)

(3)

(4)

(5)

3. 改进的浸入界面方法求解一维椭圆问题

3.1. 含一个间断节点情形的差分格式构造

(6)

(7)

(8)

(9)

(10)

(11)

3.2. 含两个间断节点情形的差分格式构造

(12)

(13)

1)

2)

3)

1)

(14)

(15)

(16)

(17)

2)

，在点处的泰勒展式，再结合上述一个节点的情况，和连接条件(4) 进而得到的关系，用以下形式表出

(18)

(19)

(20)

3)

(21)

(22)

(23)

4. 改进的浸入界面方法求解一维热传导问题

(24)

(25)

Dirichlet边界条件

(26)

(27)

，连续的扩散方程的Crank-Nickson格式为

(28)

(29)

(30)

(31)

(32)

(33)

5. 数值算例

5.1. 算例1

(34)

，步长，利用改进的浸入界面方法求解。数值结果见表1和图1，表明当函数为分段二次函数时，采用浸入界面方法可以精确的得到该问题的解，误差为机器误差。该算例的截断误差如表1。这里是无穷范数误差，是L2范数误差，当改进的浸入界面方法求解此问题时，得到的误差为机器误差，因此随着剖分加密，机器误差的累积使得剖分越密，误差越大。

Table 1. Two types of errors for example 1

Figure 1. The error distribution for example 1

5.2. 算例2

(35)

5.3. 算例3

Table 2. Two types of error for example 2

Table 3. The errors and orders for example 3

Figure 2. The error distribution for example 2

Figure 3. The error of example 3

5.4. 算例4

Table 4. The errors and orders for example 4

Figure 4. The error of example 4

6. 结论及创新点

1) 首先求解此类问题时，我们了解到李志林教授所提出的方法即在点处进行展开差分，但考虑到拟合精度的优良性，我们对其差分格式进行了修改。即我们主要是在点处进行泰勒展开。将其精度调至三阶。并对其进行了算例检验。2) 基于上面我们所推导方法，对其运用在了热传导问题中。

Immersed Interface Method for Elliptic Interface and Heat Conduction Problem. 应用数学进展,02,136-149. doi: 10.12677/AAM.2015.42019

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