﻿ 变系数椭圆型方程定解问题的一种数值解法 A Numerical Solution to a Solution Problem for Elliptic Equation with Variable Coefficients

Vol. 07  No. 10 ( 2018 ), Article ID: 27253 , 9 pages
10.12677/AAM.2018.710151

A Numerical Solution to a Solution Problem for Elliptic Equation with Variable Coefficients

Duowei Zhu1, Adil Nazakat1, Mamat Imam2, Abduwali Abdirixit1

1College of Mathematics and Systems Science, Xinjiang University, Urumqi Xinjiang

2College of Mathematics and Information, Shaanxi Normal University, Xi’an Shaanxi

Received: Oct. 1st, 2018; accepted: Oct. 17th, 2018; published: Oct. 24th, 2018

ABSTRACT

This paper proposes a numerical solution for solving that problem of fixed solution of the variable coefficient elliptic equation, and we get the corresponding error analysis, the method is verified by numerical experiments; convergence speed and small error, in time and space can achieve second-order accuracy.

Keywords:Variable Coefficient Elliptic Equation, Numerical Solution, The Error Analysis

1新疆大学，数学与系统科学学院，新疆 乌鲁木齐

2陕西师范大学，数学与信息学院，陕西 西安

1. 引言

$\left\{\begin{array}{l}-\underset{i=1}{\overset{2}{\sum }}\underset{j=1}{\overset{2}{\sum }}\frac{\partial }{\partial {x}_{i}}\left({a}_{ij}\left({x}_{1},{x}_{2}\right)\frac{\partial u\left({x}_{1},{x}_{2}\right)}{\partial {x}_{j}}\right)=f\left({x}_{i},{x}_{j}\right),\forall \left({x}_{1},{x}_{2}\right)\in \Omega ,\\ {u\left({x}_{1},{x}_{2}\right)|}_{\Gamma }=g\left({x}_{1},{x}_{2}\right),\forall \left({x}_{1},{x}_{2}\right)\in \Gamma .\end{array}$ (1)

2. 差分格式的建立

${x}_{1i}=ih,i=0,1,2,\cdots ,m,$

${x}_{2j}=jk,j=0,1,2,\cdots n,$

${\Gamma }_{h}$ 表示网格线 ${x}_{1}={x}_{1i}$${x}_{2}={x}_{2j}$$\Gamma$ 的交点的集合，并称此类点为边界节点。

${\Omega }_{h}$ 上的网格函数为：

$U=\left\{{U}_{ij}|0\le i\le m,0\le j\le n\right\}$

${U}_{ij}=\left\{u\left({x}_{1i},{x}_{2j}\right)|0\le i\le m,0\le j\le n\right\}.$

2.1. 交错五点差分格式

${\left[\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{\partial r}\right)\right]}_{ij}\approx \frac{1}{{r}_{i}{\left(\Delta r\right)}^{2}}\left[{r}_{i+\frac{1}{2}}u\left({r}_{i+1,{\theta }_{j}}\right)-\left({r}_{i+\frac{1}{2}}+{r}_{i-\frac{1}{2}}\right)u\left({r}_{i,{\theta }_{j}}\right)+{r}_{i-\frac{1}{2}}u\left({r}_{i-1,{\theta }_{j}}\right)\right]$ (2)

$\frac{\left({a}_{ij}\left({x}_{1},{x}_{2}\right)\frac{\partial u\left({x}_{1},{x}_{2}\right)}{\partial {x}_{j}}\right)|{}_{\left(i+\frac{1}{2},j\right)}-\left({a}_{ij}\left({x}_{1},{x}_{2}\right)\frac{\partial u\left({x}_{1},{x}_{2}\right)}{\partial {x}_{j}}\right)|{}_{\left(i-\frac{1}{2},j\right)}}{h},$

$\frac{\partial }{\partial {x}_{1}}\left({a}_{11}\frac{\partial u}{\partial {x}_{1}}\right)=\frac{{a}_{i+\frac{1}{2},j}^{11}\left({u}_{i+1,j}-{u}_{i,j}\right)-{a}_{i-\frac{1}{2},j}^{11}\left({u}_{i,j}-{u}_{i-1,j}\right)}{{h}^{2}}+O\left({h}^{2}\right),$ (3)

$\frac{\partial }{\partial {x}_{1}}\left({a}_{12}\frac{\partial u}{\partial {x}_{2}}\right)=\frac{{a}_{i+\frac{1}{2},j}^{12}\left({u}_{i+\frac{1}{2},j+\frac{1}{2}}-{u}_{i+\frac{1}{2},j-\frac{1}{2}}\right)-{a}_{i-\frac{1}{2},j}^{12}\left({u}_{i-\frac{1}{2},j+\frac{1}{2}}-{u}_{i-\frac{1}{2},j-\frac{1}{2}}\right)}{{h}^{2}}+O\left(kh\right),$ (4)

$\frac{\partial }{\partial x{}_{2}}\left({a}_{21}\frac{\partial u}{\partial {x}_{1}}\right)=\frac{{a}_{i,j+\frac{1}{2}}^{21}\left({u}_{i+\frac{1}{2},j+\frac{1}{2}}-{u}_{i-\frac{1}{2},j+\frac{1}{2}}\right)-{a}_{i,j-\frac{1}{2}}^{21}\left({u}_{i+\frac{1}{2},j-\frac{1}{2}}-{u}_{i-\frac{1}{2},j-\frac{1}{2}}\right)}{{h}^{2}}+O\left(kh\right),$ (5)

$\frac{\partial }{\partial {x}_{2}}\left({a}_{22}\frac{\partial u}{\partial {x}_{2}}\right)=\frac{{a}_{i,j+\frac{1}{2}}^{22}\left({u}_{i,j+1}-{u}_{i,j}\right)-{a}_{i,j-\frac{1}{2}}^{22}\left({u}_{i,j}-{u}_{i,j-1}\right)}{{k}^{2}}+O\left({k}^{2}\right),$ (6)

${a}_{i+\frac{1}{2},j}^{11},{a}_{i+\frac{1}{2},j}^{12},{a}_{i,j+\frac{1}{2}}^{21},{a}_{i,j+\frac{1}{2}}^{22},{a}_{i-\frac{1}{2},j}^{11},{a}_{i-\frac{1}{2},j}^{12},{a}_{i,j-\frac{1}{2}}^{21},{a}_{i,j-\frac{1}{2}}^{22}.$

$\begin{array}{l}-\frac{{a}_{i+\frac{1}{2},j}^{11}}{{h}^{2}}{u}_{i+1,j}+\left(\frac{{a}_{i+\frac{1}{2},j}^{11}+{a}_{i-\frac{1}{2},j}^{11}}{{h}^{2}}+\frac{{a}_{i,j+\frac{1}{2}}^{22}+{a}_{i,j-\frac{1}{2}}^{22}}{{k}^{2}}\right){u}_{i,j}-\frac{{a}_{i-\frac{1}{2},j}^{11}}{{h}^{2}}{u}_{i-1,j}-\frac{{a}_{i,j+\frac{1}{2}}^{22}}{{k}^{2}}{u}_{i,j+1}\\ -\frac{{a}_{i,j-\frac{1}{2}}^{22}}{{k}^{2}}{u}_{i,j-1}-\left(\frac{{a}_{i+\frac{1}{2},j}^{12}+{a}_{i,j+\frac{1}{2}}^{21}}{kh}\right){u}_{i+\frac{1}{2},j+\frac{1}{2}}+\left(\frac{{a}_{i+\frac{1}{2},j}^{12}+{a}_{i,j-\frac{1}{2}}^{21}}{kh}\right){u}_{i+\frac{1}{2},j-\frac{1}{2}}\\ +\left(\frac{{a}_{i-\frac{1}{2},j}^{12}+{a}_{i,j+\frac{1}{2}}^{21}}{kh}\right){u}_{i-\frac{1}{2},j+\frac{1}{2}}-\left(\frac{{a}_{i-\frac{1}{2},j}^{12}+{a}_{i,j-\frac{1}{2}}^{21}}{kh}\right){u}_{i-\frac{1}{2},j-\frac{1}{2}}={f}_{i,j}\end{array}$ (7)

$\left(\left(i-\frac{1}{2},j-\frac{1}{2}\right),\left(i-\frac{1}{2},j+\frac{1}{2}\right),\left(i+\frac{1}{2},j-\frac{1}{2}\right),\left(i+\frac{1}{2},j+\frac{1}{2}\right)\right)$

(称为网格的中心点)交错得到的，从节点的角度看，用了五个节点 ${x}_{i,j+1},{x}_{i,j-1},{x}_{i,j},{x}_{i-1,j},{x}_{i+1,j}$ ，但是计算的时候我们用到了网格的中心点，又可以看作是九个点的差分格式。因此，我们称差分格式(7)为交错五点差分格式。

2.2. 五点差分格式

${u}_{i+\frac{1}{2},j+\frac{1}{2}}=\frac{{u}_{i+1,j}+{u}_{i,j+1}}{2}+O\left({h}^{2}+{k}^{2}\right),$ (8)

${u}_{i+\frac{1}{2},j-\frac{1}{2}}=\frac{{u}_{i+1,j}+{u}_{i,j-1}}{2}+O\left({h}^{2}+{k}^{2}\right),$ (9)

${u}_{i-\frac{1}{2},j+\frac{1}{2}}=\frac{{u}_{i-1,j}+{u}_{i,j+1}}{2}+O\left({h}^{2}+{k}^{2}\right),$ (10)

${u}_{i-\frac{1}{2},j-\frac{1}{2}}=\frac{{u}_{i-1,j}+{u}_{i,j-1}}{2}+O\left({h}^{2}+{k}^{2}\right),$ (11)

$\begin{array}{l}\left(-\frac{{a}_{i+\frac{1}{2},j}^{11}}{{h}^{2}}+\frac{{a}_{i,j-\frac{1}{2}}^{21}-{a}_{i,j+\frac{1}{2}}^{21}}{2kh}\right){u}_{i+1,j}+\left(\frac{{a}_{i+\frac{1}{2},j}^{11}+{a}_{i-\frac{1}{2},j}^{11}}{{h}^{2}}+\frac{{a}_{i,j+\frac{1}{2}}^{22}+{a}_{i,j-\frac{1}{2}}^{22}}{{k}^{2}}\right){u}_{i,j}+\left(-\frac{{a}_{i-\frac{1}{2},j}^{11}}{{h}^{2}}+\frac{{a}_{i,j+\frac{1}{2}}^{21}-{a}_{i,j-\frac{1}{2}}^{21}}{2kh}\right){u}_{i-1,j}\\ +\left(-\frac{{a}_{i,j+\frac{1}{2}}^{22}}{{k}^{2}}+\frac{{a}_{i-\frac{1}{2},j}^{12}-{a}_{i+\frac{1}{2},j}^{12}}{2kh}\right){u}_{i,j+1}+\left(-\frac{{a}_{i,j-\frac{1}{2}}^{22}}{{k}^{2}}+\frac{{a}_{i+\frac{1}{2},j}^{12}-{a}_{i-\frac{1}{2},j}^{12}}{2kh}\right){u}_{i,j-1}={f}_{i,j}\end{array}$ (12)

2.3. 九点差分格式

$\frac{\partial }{\partial {x}_{1}}\left({a}_{11}\frac{\partial u}{\partial {x}_{1}}\right)=\frac{{a}_{i+\frac{1}{2},j}^{11}\left({u}_{i+1,j}-{u}_{i,j}\right)-{a}_{i-\frac{1}{2},j}^{11}\left({u}_{i,j}-{u}_{i-1,j}\right)}{{h}^{2}}+O\left({h}^{2}\right),$ (13)

$\frac{\partial }{\partial {x}_{1}}\left({a}_{12}\frac{\partial u}{\partial {x}_{2}}\right)=\frac{{a}_{i+1,j}^{12}\left({u}_{i+1,j+1}-{u}_{i+1,j-1}\right)-{a}_{i-1,j}^{12}\left({u}_{i-1,j+1}-{u}_{i-1,j-1}\right)}{4kh}+O\left(kh\right),$ (14)

$\frac{\partial }{\partial {x}_{2}}\left({a}_{21}\frac{\partial u}{\partial {x}_{1}}\right)=\frac{{a}_{i,j+1}^{21}\left({u}_{i+1,j+1}-{u}_{i-1,j+1}\right)-{a}_{i,j-1}^{21}\left({u}_{i+1,j-1}-{u}_{i-1,j-1}\right)}{4kh}+O\left(kh\right),$ (15)

$\frac{\partial }{\partial {x}_{2}}\left({a}_{22}\frac{\partial u}{\partial {x}_{2}}\right)=\frac{{a}_{i,j+\frac{1}{2}}^{22}\left({u}_{i,j+1}-{u}_{i,j}\right)-{a}_{i,j-\frac{1}{2}}^{22}\left({u}_{i,j}-{u}_{i,j-1}\right)}{{h}^{2}}+O\left({k}^{2}\right),$ (16)

$\begin{array}{l}-\frac{{a}_{i+\frac{1}{2},j}^{11}}{{h}^{2}}{u}_{i+1,j}+\left(\frac{{a}_{i+\frac{1}{2},j}^{11}+{a}_{i-\frac{1}{2},j}^{11}}{{h}^{2}}+\frac{{a}_{i,j+\frac{1}{2}}^{22}+{a}_{i,j-\frac{1}{2}}^{22}}{{k}^{2}}\right){u}_{i,j}-\frac{{a}_{i-\frac{1}{2},j}^{11}}{{h}^{2}}{u}_{i-1,j}-\frac{{a}_{i,j-\frac{1}{2}}^{22}}{{k}^{2}}{u}_{i,j-1}\\ -\frac{{a}_{i,j+\frac{1}{2}}^{22}}{{k}^{2}}{u}_{i,j+1}-\left(\frac{{a}_{i+1,j}^{12}+{a}_{i,j+1}^{21}}{4kh}\right){u}_{i+1,j+1}+\left(\frac{{a}_{i+1,j}^{12}+{a}_{i,j-1}^{21}}{4kh}\right){u}_{i+1,j-1}\\ +\left(\frac{{a}_{i-1,j}^{12}+{a}_{i,j+1}^{21}}{4kh}\right){u}_{i-1,j+1}-\left(\frac{{a}_{i-1,j}^{12}+{a}_{i,j-1}^{21}}{4kh}\right){u}_{i-1,j-1}={f}_{i,j}\end{array}$ (17)

3. 差分格式的误差分析

$\begin{array}{c}T\left({x}_{1},{x}_{2}\right)=-\frac{{h}^{2}}{6}\frac{\partial {a}_{11}}{\partial {x}_{1}}\frac{{\partial }^{3}u}{\partial {x}_{1}{}^{3}}\left(\xi ,{x}_{2}\right)-\frac{{k}^{2}}{6}\frac{\partial {a}_{22}}{\partial {x}_{2}}\frac{{\partial }^{3}u}{\partial {x}_{2}{}^{3}}\left({x}_{1},\eta \right)-\frac{{h}^{2}}{24}\frac{\partial {a}_{21}}{\partial {x}_{2}}\frac{{\partial }^{3}u}{\partial {x}_{1}{}^{3}}\left(\xi ,{x}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{k}^{2}}{24}\frac{\partial {a}_{12}}{\partial {x}_{1}}\frac{{\partial }^{3}u}{\partial {x}_{2}{}^{3}}\left({x}_{1},\eta \right)-\frac{{h}^{2}}{8}\frac{\partial {a}_{12}}{\partial {x}_{1}}\frac{{\partial }^{2}u}{\partial {x}_{1}{}^{2}}\left(\xi ,{x}_{2}\right)\frac{\partial u}{\partial {x}_{2}}\left({x}_{1},\eta \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{k}^{2}}{8}\frac{\partial {a}_{21}}{\partial {x}_{2}}\frac{{\partial }^{2}u}{\partial {x}_{2}{}^{2}}\left({x}_{1},\eta \right)\frac{\partial u}{\partial {x}_{1}}\left(\xi ,{x}_{2}\right)\end{array}$ (18)

$T\left({x}_{1},{x}_{2}\right)=-\frac{{h}^{2}}{6}\left(\frac{\partial {a}_{11}}{\partial {x}_{1}}+\frac{\partial {a}_{21}}{\partial {x}_{2}}\right)\frac{{\partial }^{3}u}{\partial {x}_{1}{}^{3}}\left(\xi ,{x}_{2}\right)-\frac{{k}^{2}}{6}\left(\frac{\partial {a}_{22}}{\partial {x}_{2}}+\frac{\partial {a}_{12}}{\partial {x}_{1}}\right)\frac{{\partial }^{3}u}{\partial {x}_{2}{}^{3}}\left({x}_{1},\eta \right)$ (19)

$\begin{array}{c}T\left({x}_{1},{x}_{2}\right)=-\frac{{h}^{2}}{6}\frac{\partial {a}_{11}}{\partial {x}_{1}}\frac{{\partial }^{3}u}{\partial {x}_{1}{}^{3}}\left(\xi ,{x}_{2}\right)-\frac{{k}^{2}}{6}\frac{\partial {a}_{22}}{\partial {x}_{2}}\frac{{\partial }^{3}u}{\partial {x}_{2}{}^{3}}\left({x}_{1},\eta \right)-\frac{{h}^{2}}{12}\frac{\partial {a}_{21}}{\partial {x}_{2}}\frac{{\partial }^{3}u}{\partial {x}_{1}{}^{3}}\left(\xi ,{x}_{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{k}^{2}}{12}\frac{\partial {a}_{12}}{\partial {x}_{1}}\frac{{\partial }^{3}u}{\partial {x}_{2}{}^{3}}\left({x}_{1},\eta \right)-\frac{{h}^{2}}{12}\frac{\partial {a}_{12}}{\partial {x}_{1}}\frac{{\partial }^{2}u}{\partial {x}_{1}{}^{2}}\left(\xi ,{x}_{2}\right)\frac{\partial u}{\partial {x}_{2}}\left({x}_{1},\eta \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{k}^{2}}{12}\frac{\partial {a}_{21}}{\partial {x}_{2}}\frac{{\partial }^{2}u}{\partial {x}_{2}{}^{2}}\left({x}_{1},\eta \right)\frac{\partial u}{\partial {x}_{1}}\left(\xi ,{x}_{2}\right)\end{array}$ (20)

4. 数值例子

$\left\{\begin{array}{l}-\underset{i=1}{\overset{2}{\sum }}\underset{j=1}{\overset{2}{\sum }}\frac{\partial }{\partial {x}_{1}}\left({a}_{i,j}\left({x}_{1},{x}_{2}\right)\frac{\partial u\left({x}_{1},{x}_{2}\right)}{\partial {x}_{j}}\right)=f\left({x}_{1},{x}_{2}\right),\forall \left({x}_{1},{x}_{2}\right)\in \Omega ,\\ {a}_{i,j}\left({x}_{1},{x}_{2}\right)={x}_{i}{x}_{j},i,j=1,2,\\ f\left({x}_{1},{x}_{2}\right)=-3{x}_{1}\mathrm{cos}{x}_{1}\mathrm{cos}{x}_{2}+{x}_{1}^{2}\mathrm{sin}{x}_{1}\mathrm{cos}{x}_{2}+3{x}_{2}\mathrm{sin}{x}_{1}\mathrm{sin}{x}_{2}+2{x}_{1}{x}_{2}\mathrm{cos}{x}_{1}\mathrm{sin}{x}_{2}+{x}_{2}^{2}\mathrm{sin}{x}_{1}\mathrm{cos}{x}_{2},\\ {u\left({x}_{1},{x}_{2}\right)|}_{\Gamma }=0,\left({x}_{1},{x}_{2}\right)\in \partial \Omega \end{array}$ (21)

Table 1. Exact solution and numerical solution of three difference schemes

$M=N=32$ ，即 $h=k=\frac{1}{32}$ ，用上述的三种差分格式解问题(1)，得到的准确解与数值解的图像如图1

$M=N=32$ ，即 $h=k=\frac{1}{32}$ ，用上述的三种差分格式求解问题(1)，得到准确解与数值解的误差如图2

Figure 1. M = N = 32, Exact solution and numerical solution of three difference schemes

Table 2. The number of iterations calculated by Gauss-Seidel and SOR iterative methods in three difference schemes

Table 3. Error and error order of three difference schemes

Figure 2. Error of three difference schemes

5. 结论

A Numerical Solution to a Solution Problem for Elliptic Equation with Variable Coefficients[J]. 应用数学进展, 2018, 07(10): 1299-1307. https://doi.org/10.12677/AAM.2018.710151

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