二维椭圆边值问题的差分格式.docx
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二维椭圆边值问题的差分格式
二维椭圆边值问题的差分格式
一．问题介绍
考虑Poisson方程：
（1）
G是xy平面上一有界区域，其边界
为分段光滑曲线。
在
上u满足下列边值条件之一：
（2）
（第一边值条件），
（3）
（第二边值条件），
（4）
（第三边值条件），
及
都是连续函数，k
。
本节讨论逼近方程
（1）及相应边值条件的差分格式。
二．区域剖分
取定沿x轴和y轴方向的步长
和
，
。
作两族与坐标轴平行的直线：
两族直线的交点
称为网点或节点，记为
或（i,j）。
说两个节点
和
是相邻的，如果
或
。
以
表示所有属于G内部的节点集合，并称如此的节点为内点。
以
表示网线
或
与
的交点集合，并称如此的点为界点。
令
，则
就是代替域
的网点集合。
若内点
的四个相邻点都属于
，就称为正则内点；否则称为非正则内点。
三．离散格式
1.五点差分格式
假定
为正则内点。
沿x，y方向分别用二阶中心差商代替
和
，则有差分方程：
[
+
]=
，
式中
表示节点（i,j）上的函数值。
2.九点差分格式
利用Taylor展式，将
和
在
处展开，然后相加化简就得到逼近Poisson方程的九点差分格式：
四．格式稳定性
1.五点差分格式的收敛阶为
2.九点差分格式的收敛阶为
五．数值例子
例1令u（x,y）=sin（
x）sin（
y），区间[0,1]。
程序结果如下：
输入划分区间的点数n（输入0结束程序）：
5
输入划分区间的点数m（输入0结束程序）：
5
xiyj准确值u（x,y）近似值u[i][j]误差err[i]
0.1666670.1666670.2500000.2557910.005791
0.1666670.3333330.4330130.4430430.010030
0.1666670.5000000.5000000.5115810.011581
0.1666670.6666670.4330130.4430430.010030
0.1666670.8333330.2500000.2557910.005791
0.3333330.1666670.4330130.4430430.010030
0.3333330.3333330.7500000.7673720.017372
0.3333330.5000000.8660250.8860850.020060
0.3333330.6666670.7500000.7673720.017372
0.3333330.8333330.4330130.4430430.010030
0.5000000.1666670.5000000.5115810.011581
0.5000000.3333330.8660250.8860850.020060
0.5000000.5000001.0000001.0231630.023163
0.5000000.6666670.8660250.8860850.020060
0.5000000.8333330.5000000.5115810.011581
0.6666670.1666670.4330130.4430430.010030
0.6666670.3333330.7500000.7673720.017372
0.6666670.5000000.8660250.8860850.020060
0.6666670.6666670.7500000.7673720.017372
0.6666670.8333330.4330130.4430430.010030
0.8333330.1666670.2500000.2557910.005791
0.8333330.3333330.4330130.4430430.010030
0.8333330.5000000.5000000.5115810.011581
0.8333330.6666670.4330130.4430430.010030
0.8333330.8333330.2500000.2557910.005791
误差与步长的2-范数e[0]：
0.000805
n为5和m为5时其最大误差：
0.023163
输入划分区间的点数n（输入0结束程序）：
10
输入划分区间的点数m（输入0结束程序）：
10
xiyj准确值u（x,y）近似值u[i][j]误差err[i]
0.0909090.0909090.0793730.0799150.000542
0.0909090.1818180.1523160.1533560.001040
0.0909090.2727270.2129190.2143720.001453
0.0909090.3636360.2562730.2580220.001749
0.0909090.4545450.2788650.2807680.001903
0.0909090.5454550.2788650.2807680.001903
0.0909090.6363640.2562730.2580220.001749
0.0909090.7272730.2129190.2143720.001453
0.0909090.8181820.1523160.1533560.001040
0.0909090.9090910.0793730.0799150.000542
0.1818180.0909090.1523160.1533560.001040
0.1818180.1818180.2922920.2942870.001995
0.1818180.2727270.4085890.4113780.002789
0.1818180.3636360.4917840.4951410.003356
0.1818180.4545450.5351380.5387900.003652
0.1818180.5454550.5351380.5387900.003652
0.1818180.6363640.4917840.4951410.003356
0.1818180.7272730.4085890.4113780.002789
0.1818180.8181820.2922930.2942870.001995
0.1818180.9090910.1523160.1533560.001040
0.2727270.0909090.2129190.2143720.001453
0.2727270.1818180.4085890.4113780.002789
0.2727270.2727270.5711570.5750560.003898
0.2727270.3636360.6874540.6921460.004692
0.2727270.4545450.7480570.7531630.005106
0.2727270.5454550.7480570.7531630.005106
0.2727270.6363640.6874540.6921460.004692
0.2727270.7272730.5711570.5750560.003898
0.2727270.8181820.4085890.4113780.002789
0.2727270.9090910.2129190.2143720.001453
0.3636360.0909090.2562730.2580220.001749
0.3636360.1818180.4917840.4951410.003356
0.3636360.2727270.6874540.6921460.004692
0.3636360.3636360.8274300.8330780.005647
0.3636360.4545450.9003730.9065180.006145
0.3636360.5454550.9003730.9065180.006145
0.3636360.6363640.8274300.8330780.005647
0.3636360.7272730.6874540.6921460.004692
0.3636360.8181820.4917840.4951410.003356
0.3636360.9090910.2562730.2580220.001749
0.4545450.0909090.2788650.2807680.001903
0.4545450.1818180.5351380.5387900.003652
0.4545450.2727270.7480570.7531630.005106
0.4545450.3636360.9003730.9065180.006145
0.4545450.4545450.9797460.9864330.006687
0.4545450.5454550.9797460.9864330.006687
0.4545450.6363640.9003730.9065180.006145
0.4545450.7272730.7480570.7531630.005106
0.4545450.8181820.5351380.5387900.003652
0.4545450.9090910.2788650.2807680.001903
0.5454550.0909090.2788650.2807680.001903
0.5454550.1818180.5351380.5387900.003652
0.5454550.2727270.7480570.7531630.005106
0.5454550.3636360.9003730.9065180.006145
0.5454550.4545450.9797460.9864330.006687
0.5454550.5454550.9797460.9864330.006687
0.5454550.6363640.9003730.9065180.006145
0.5454550.7272730.7480570.7531630.005106
0.5454550.8181820.5351380.5387900.003652
0.5454550.9090910.2788650.2807680.001903
0.6363640.0909090.2562730.2580220.001749
0.6363640.1818180.4917840.4951410.003356
0.6363640.2727270.6874540.6921460.004692
0.6363640.3636360.8274300.8330780.005647
0.6363640.4545450.9003730.9065180.006145
0.6363640.5454550.9003730.9065180.006145
0.6363640.6363640.8274300.8330780.005647
0.6363640.7272730.6874540.6921460.004692
0.6363640.8181820.4917840.4951410.003356
0.6363640.9090910.2562730.2580220.001749
0.7272730.0909090.2129190.2143720.001453
0.7272730.1818180.4085890.4113780.002789
0.7272730.2727270.5711570.5750560.003898
0.7272730.3636360.6874540.6921460.004692
0.7272730.4545450.7480570.7531630.005106
0.7272730.5454550.7480570.7531630.005106
0.7272730.6363640.6874540.6921460.004692
0.7272730.7272730.5711570.5750560.003898
0.7272730.8181820.4085890.4113780.002789
0.7272730.9090910.2129190.2143720.001453
0.8181820.0909090.1523160.1533560.001040
0.8181820.1818180.2922930.2942870.001995
0.8181820.2727270.4085890.4113780.002789
0.8181820.3636360.4917840.4951410.003356
0.8181820.4545450.5351380.5387900.003652
0.8181820.5454550.5351380.5387900.003652
0.8181820.6363640.4917840.4951410.003356
0.8181820.7272730.4085890.4113780.002789
0.8181820.8181820.2922930.2942870.001995
0.8181820.9090910.1523160.1533560.001040
0.9090910.0909090.0793730.0799150.000542
0.9090910.1818180.1523160.1533560.001040
0.9090910.2727270.2129190.2143720.001453
0.9090910.3636360.2562730.2580220.001749
0.9090910.4545450.2788650.2807680.001903
0.9090910.5454550.2788650.2807680.001903
0.9090910.6363640.2562730.2580220.001749
0.9090910.7272730.2129190.2143720.001453
0.9090910.8181820.1523160.1533560.001040
0.9090910.9090910.0793730.0799150.000542
误差与步长的2-范数e[1]：
0.000128
那么log（e[1]/e[0]）的值是：
1.837767
n为10和m为10时其最大误差：
0.006687
以上数值结果是通过五点差分格式用C++工具求得的；
下面采用九点差分格式用C++工具求数值结果。
输入划分区间的点数n（输入0结束程序）：
5
输入划分区间的点数m（输入0结束程序）：
5
xiyj准确值u（x,y）近似值u[i][j]误差err[i]
0.1666670.1666670.2500000.2496780.000322
0.1666670.3333330.4330130.4324550.000558
0.1666670.5000000.5000000.4993560.000644
0.1666670.6666670.4330130.4324550.000558
0.1666670.8333330.2500000.2496780.000322
0.3333330.1666670.4330130.4324550.000558
0.3333330.3333330.7500000.7490340.000966
0.3333330.5000000.8660250.8649100.001115
0.3333330.6666670.7500000.7490340.000966
0.3333330.8333330.4330130.4324550.000558
0.5000000.1666670.5000000.4993560.000644
0.5000000.3333330.8660250.8649100.001115
0.5000000.5000001.0000000.9987120.001288
0.5000000.6666670.8660250.8649100.001115
0.5000000.8333330.5000000.4993560.000644
0.6666670.1666670.4330130.4324550.000558
0.6666670.3333330.7500000.7490340.000966
0.6666670.5000000.8660250.8649100.001115
0.6666670.6666670.7500000.7490340.000966
0.6666670.8333330.4330130.4324550.000558
0.8333330.1666670.2500000.2496780.000322
0.8333330.3333330.4330130.4324550.000558
0.8333330.5000000.5000000.4993560.000644
0.8333330.6666670.4330130.4324550.000558
0.8333330.8333330.2500000.2496780.000322
误差与步长的2-范数e[0]：
0.000002
n为5和m为5时其最大误差：
0.001288
输入划分区间的点数n（输入0结束程序）：
10
输入划分区间的点数m（输入0结束程序）：
10
xiyj准确值u（x,y）近似值u[i][j]误差err[i]
0.0909090.0909090.0793730.0793640.000009
0.0909090.1818180.1523160.1522990.000017
0.0909090.2727270.2129190.2128950.000024
0.0909090.3636360.2562730.2562440.000029
0.0909090.4545450.2788650.2788340.000031
0.0909090.5454550.2788650.2788340.000031
0.0909090.6363640.2562730.2562440.000029
0.0909090.7272730.2129190.2128950.000024
0.0909090.8181820.1523160.1522990.000017
0.0909090.9090910.0793730.0793640.000009
0.1818180.0909090.1523160.1522990.000017
0.1818180.1818180.2922920.2922600.000033
0.1818180.2727270.4085890.4085430.000046
0.1818180.3636360.4917840.4917290.000055
0.1818180.4545450.5351380.5350780.000060
0.1818180.5454550.5351380.5350780.000060
0.1818180.6363640.4917840.4917290.000055
0.1818180.7272730.4085890.4085430.000046
0.1818180.8181820.2922930.2922600.000033
0.1818180.9090910.1523160.1522990.000017
0.2727270.0909090.2129190.2128950.000024
0.2727270.1818180.4085890.4085430.000046
0.2727270.2727270.5711570.5710940.000064
0.2727270.3636360.6874540.6873770.000077
0.2727270.4545450.7480570.7479730.000084
0.2727270.5454550.7480570.7479730.000084
0.2727270.6363640.6874540.6873770.000077
0.2727270.7272730.5711570.5710940.000064
0.2727270.8181820.4085890.4085430.000046
0.2727270.9090910.2129190.2128950.000024
0.3636360.0909090.2562730.2562440.000029
0.3636360.1818180.4917840.4917290.000055
0.3636360.2727270.6874540.6873770.000077
0.3636360.3636360.8274300.8273380.000093
0.3636360.4545450.9003730.9002730.000101
0.3636360.5454550.9003730.9002730.000101
0.3636360.6363640.8274300.8273380.000093
0.3636360.7272730.6874540.6873770.000077
0.3636360.8181820.4917840.4917290.000055
0.3636360.
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