偏微分课程课件9 椭圆型方程的有限差分方法I

1、1,第五章,椭圆型方程的差分方法,一,Poisson,方程,二）差分格式的性质,三）边界条件的处理,四）变系数方程,五）双调和方程,六）特征值问题,2,一,Poisson,方程,2,2,2,2,u,u,u,f,x,y,x,y,D,x,y,0,0,D,x,y,x,a,y,b,0,0,0,0,D,x,y,x,a,y,b,y,b,x,a,区域以及边界离散,1,1,a,b,h,k,I,J,1,1,i,j,i,h,j,x,y,x,ih,i,i,I,D,y,jk,j,j,J,内点为,0,1,0,1,1,0,1,0,1,1,i,j,i,j,h,x,y,x,ih,y,jk,D,i,I,j,J,J,j,J,i,

2、I,I,L,L,边界点为,4,x,y,a,b,0,0,u,u,x,y,x,y,a,b,内点,边界点,返回,2,2,I,J,总,5,x,y,a,b,0,0,u,u,x,y,x,y,a,b,返回,1,2,I,u,u,u,L,1,2,2,I,I,I,u,u,u,L,IJ,u,L,I,J,内点,1,五点差分格式,利用,Taylor,级数展开有,2,2,2,4,4,1,2,2,4,4,1,1,2,1,1,2,24,i,j,i,j,i,j,j,j,ij,i,i,u,x,h,y,u,x,y,u,x,h,y,h,u,h,u,y,u,y,x,x,x,x,x,2,2,2,4,4,1,2,2,4,4,1,1,2,1

3、,1,2,24,i,j,i,j,i,j,i,i,ij,j,j,u,x,y,k,u,x,y,u,x,y,k,k,u,h,u,x,u,x,y,y,x,y,y,2,2,2,2,u,u,Lu,u,f,x,y,x,y,1,1,1,1,2,2,2,2,i,j,ij,i,j,i,j,ij,i,j,h,ij,h,ij,ij,u,u,u,u,u,u,L,u,u,f,h,k,ij,i,j,f,f,x,y,其中,则,2,2,i,j,h,i,j,i,j,T,x,y,L,u,Lu,O,h,k,8,i,j,由于上差分方程中,只出现,u,在,i,j,及其四个,临点上的值,故称为五,点差分格式,1,1,1,1,2,2,2,2

4、,i,j,ij,i,j,i,j,ij,i,j,h,ij,ij,u,u,u,u,u,u,u,f,h,k,h,ij,ij,i,j,h,ij,ij,i,j,h,ij,i,j,i,j,ij,u,f,x,y,D,u,x,y,D,u,u,x,y,x,y,x,y,D,其中,Possion,方程第一边值问题的差分逼近为,改写为代数方程,仅考虑,1,1,1,1,2,2,2,2,i,j,i,j,i,j,i,j,i,j,i,j,i,j,u,u,u,u,u,u,f,h,k,1,1,i,I,j,J,L,L,h,k,解,2,2,2,2,2,2,0,log,1,0,1,u,u,x,y,D,x,y,u,x,y,x,y,x,y

5、,D,D,x,y,x,y,例：五差分格式求解,1,1,1,1,2,1,4,i,j,i,j,i,j,i,j,i,j,i,j,u,u,u,u,u,f,h,先仅下标,j,小的未知量放前面,相同的,j,的再按,i,从小到大顺利排列,11,1,1,i,J,j,I,L,L,按自然顺序排列网点,i,j,1,1,1,1,2,j,j,J,i,I,i,I,I,j,i,L,L,L,L,定义向量,1,2,2,1,1,1,1,J,I,I,h,J,I,u,u,u,u,u,u,u,L,L,L,L,于是差分方程为,2,1,h,Hu,g,h,12,1,1,i,I,j,J,L,L,分析系数矩阵,H,对于第一个结点,1,1,1,1

6、,1,0,0,1,2,1,1,2,1,1,2,1,4,u,u,u,u,f,h,u,4,1,0,1,0,0,L,L,1,j,时,2,j,I,个,1,1,1,1,2,1,4,i,j,i,j,i,j,i,j,i,j,i,j,u,u,u,u,u,f,h,I,个,1,1,H,对应,的第一行,未知节点,1,j,2,j,1,1,i,I,j,J,L,L,分析系数矩阵,H,对于第二个结点,2,1,2,1,2,0,1,1,3,1,2,2,2,1,2,1,4,u,u,u,u,f,h,u,4,1,0,1,0,0,1,4,1,0,1,0,L,L,L,L,1,j,2,j,I,个,1,1,1,1,2,1,4,i,j,i,j

7、,i,j,i,j,i,j,i,j,u,u,u,u,u,f,h,I,个,2,1,H,节点对应,的第二行,1,j,2,j,1,1,i,I,j,J,L,L,1,1,1,1,2,1,4,i,j,i,j,i,j,i,j,i,j,ij,u,u,u,u,u,f,h,分析系数矩阵,H,对于前,I,个结点,系数矩阵,H,为,4,1,1,1,4,1,1,1,4,1,1,1,4,1,O,O,O,O,L,L,1,j,2,j,B,I,B,I,I,B,I,H,I,B,I,I,B,O,O,O,1,1,1,1,2,1,4,i,j,i,j,ij,i,j,i,j,ij,u,u,u,u,u,f,h,五点差分格式,2,1,h,Hu,

8、g,h,等价于,B,I,为,I,维方阵,H,为,I*J,维方阵,1,1,i,I,j,J,L,L,2,2,2,2,2,2,0,0,1,0,1,lg(1,u,u,x,y,D,x,y,u,x,y,x,y,x,y,D,1,3,h,k,取,x,y,0,1,1,1,2,u,u,3,4,u,u,8,7,3,4,1,2,6,5,对于第一个结点,1,2,8,3,1,2,4,0,u,u,u,h,即,1,2,3,1,8,4,u,u,u,1,1,1,1,2,1,4,i,j,i,j,ij,i,j,i,j,ij,u,u,u,u,u,f,h,例,17,同理,对于另外三个结点,分别有,1,2,4,6,7,4,u,u,u,x,

9、y,0,1,1,1,2,u,u,3,4,u,u,8,7,3,4,1,2,6,5,1,3,4,2,3,4,u,u,u,2,4,3,4,5,4,u,u,u,联立,有线性代数方程组,1,8,1,6,7,2,2,3,3,4,5,4,4,1,1,0,1,4,0,1,1,0,4,1,0,1,1,4,u,u,u,u,10,16,lg,lg,9,9,2,2,lg(1,u,x,y,x,y,x,y,D,1,2,3,1,8,4,u,u,u,18,计算机实现,五点差分格式的系数矩阵,算法,程序,返回,19,五点差分格式的系数矩阵,对,possion,方程第一边值问题建立五点差分格式,1,1,1,1,2,1,4,1,1

10、,i,j,i,j,ij,i,j,i,j,ij,u,u,u,u,u,f,h,i,I,j,J,L,L,将双下标数列,u,i,j,按自然顺序排成单下标,u,k,那么,1,1,2,1,4,1,k,I,k,k,k,k,I,k,u,u,u,u,u,f,h,k,IJ,L,20,这样有代数方程组,Hu=g,其中,1,1,2,1,4,1,k,I,k,k,k,k,I,k,u,u,u,u,u,f,h,k,IJ,L,21,这样有代数方程组,Hu=g,其中,2,1,B,I,I,B,I,H,h,I,B,I,I,B,O,O,O,4,1,1,4,1,1,4,1,1,4,B,O,O,O,H,是稀疏矩阵,每一,行最多,5,个数不

11、为,0,H,是对角占优矩阵,H,是对称正定,1,试,求出解,Poisson,方程,u,f,x,y,的五点差分格式,1,1,1,1,2,1,4,i,j,i,j,ij,i,j,i,j,ij,u,u,u,u,u,f,h,的局部截断误差（相容性,课堂练习,23,微分方程,差分格式,真解,u,u,x,y,真解,u,u,n,实用性分析,相,容,性,收,敛,性,稳定性,椭圆型方程收敛性分析区别于双曲与抛物方程,对于双曲与抛物方程,24,2,2,2,2,2,2,2,2,i,j,h,i,j,i,j,h,i,j,i,j,i,j,i,j,i,j,i,j,i,j,i,j,i,j,i,j,R,u,L,u,Lu,u,x,

12、y,u,x,y,u,x,h,y,u,x,y,u,x,h,y,h,u,x,y,k,u,x,y,u,x,y,k,k,u,u,u,x,y,u,x,y,x,y,相容性,所以五点差分格式是相容的,4,4,2,2,4,4,4,4,1,12,i,j,h,k,u,x,y,O,h,k,x,y,i,j,i,j,f,x,y,f,2,九点差分格式,2,2,2,2,2,2,2,h,h,D,x,y,x,y,h,h,D,令,那么有,4,4,2,2,2,2,2,4,2,4,2,2,h,D,1,n,n,n,d,h,d,u,x,h,y,h,u,x,y,dx,n,dx,L,L,1,0,2,0,3,0,4,0,5,0,u,e,u,u

13、,e,u,u,e,u,u,e,u,u,e,u,等,2,3,1,1,2,3,n,x,x,x,x,x,e,n,L,L,d,h,dx,e,u,x,y,0,2,3,1,4,5,6,7,8,1,1,2,3,4,0,0,0,0,2,3,4,2,3,4,2,3,4,2,3,4,6,1,1,2,3,4,2,3,4,1,1,2,3,4,2,3,4,S,u,u,u,u,e,u,e,u,e,u,e,u,O,h,1,0,2,0,3,0,4,0,5,0,u,e,u,u,e,u,u,e,u,u,e,u,u,e,u,2,2,h,h,D,x,y,x,y,其中,0,2,3,1,4,5,6,7,8,6,2,5,6,7,8,S,u

14、,u,u,u,O,h,27,2,4,2,4,6,1,0,0,0,2,4,2,4,6,2,0,0,0,1,4,2,12,1,4,2,4,6,S,u,h,u,h,D,u,O,h,S,u,h,u,h,D,u,O,h,4,0,D,u,消去,得,2,2,4,1,2,0,0,0,2,4,20,1,6,12,S,S,u,u,h,u,O,h,h,四阶精度,0,u,f,由,2,4,1,2,0,1,4,20,6,2,S,S,u,h,f,h,f,2,0,u,f,2,2,2,2,2,1,1,3.17,tan,0,r,0,2,r,u,u,u,r,f,r,r,r,r,r,y,r,x,y,x,0,3.17,0,0,0,li

15、m,0,3.18,r,r,r,u,u,u,r,r,r,方程,的系数于,奇异，因此只当,时,有意义。为了定出有意义的解，需补充,于,有,界的条件,可设,满足,3,极坐标形式的差分格式,若求解域是圆环、环形域或扇形域,则采用极坐标是方便的，此时,Poisson,方程形如,0.5,0,1,2,1,0,1,1,2,i,j,r,r,r,i,r,i,j,h,J,j,J,L,L,和,令,1,1,1,1,1,1,2,2,2,2,2,1,1,i,j,r,i,j,i,j,i,j,i,i,i,i,i,u,r,r,r,r,r,u,r,r,r,u,r,r,u,r,r,r,任意点,用中心差商公式,2,1,1,2,2,2,

16、2,2,1,1,r,i,j,i,j,i,j,i,j,i,u,r,u,r,u,r,u,r,r,分别取等步长,0,i,j,r,i,2,2,2,1,1,r,u,u,u,r,r,r,r,r,1,1,1,1,1,1,2,2,2,2,2,1,1,2,2,3.17,1,1,2,1,i,j,ij,i,j,i,i,i,i,i,i,j,i,j,i,j,i,j,i,i,r,u,r,r,u,r,u,r,r,u,u,u,f,r,代到,就,得到逼近它的差分方程,下面用有限体积法获得在点,的,差分方程,2,2,2,1,1,3.17,r,u,u,u,r,f,r,r,r,r,r,0,j,r,0.5,0,1,2,1,0,1,1,

17、2,i,j,r,i,r,i,j,j,J,h,J,L,L,1,1,2,2,1,1,2,2,2,2,1,j,j,j,j,r,r,u,u,r,dr,d,rf,r,d,dr,r,r,r,1,1,2,2,D,r,r,r,j,j,用,r,乘方程,上积分,2,2,2,1,1,3.17,r,u,u,u,r,f,r,r,r,r,r,并对得到的方程在,1,2,1,2,1,1,2,2,1,1,2,2,1,2,1,2,2,2,1,1,2,2,1,1,1,1,j,j,j,j,j,j,j,j,r,r,r,j,j,u,u,r,dr,d,r,r,r,u,u,r,d,dr,r,r,u,u,u,u,r,r,d,u,r,u,r,dr,r,r,r,r,左,r,1,1,2,2,2,2,2,2,2,j,j,j,j,u,u,r,u,r,r,r,r,r,r,u,u,r,f,