by dr. samah mohamed mabrouk 1
TRANSCRIPT
Finite Deference Method
byDr. Samah Mohamed Mabrouk
1
www.smmabrouk.faculty.zu.edu.eg
SE301_Topic 6 Al-Amer2005 2
Difference equations for the Laplace and Poisson equations
Laplace’s equation
Poisson’s equation ),(2 yxfu
02
2
2
2
y
u
x
u
02 u
Where is the Laplacian operator
),(2
2
2
2
yxfy
u
x
u
3
Central difference approximation for Second derivative
211
2
2
)(
2
x
uuu
dx
ud iii
i
2
11
2
2
)(
2
y
uuu
dy
ud jjj
j
i+1x
y
i
x
i+1i-1 i
i-1
j+1
j
j-12
1,,1,
2
,1,,1
,
2
2
2
2
)(
2
)(
2
y
uuu
x
uuu
y
u
x
u
jijijijijiji
ji
j+1
j
y
j-1
SE301_Topic 6 Al-Amer2005 4
For x =y = h
),(
042
1,1,,1,,1
,
2
2
2
2
ji
jijijijiji
jiyxfh
uuuuu
y
u
x
u
),(
04 2
jiSNWEij yxfhuuuuu
East
West
North
South
Example 1Solve the Laplace equation on a square plate of side 12 cm, using a grid of mesh 4cm and a Dirchlet B.C. as u(x,0)=u(0,y)=u(12,y)=100 and u(x,12)=0
02
2
2
2
y
u
x
u
solution
x = 0 4 8 12 0
4
8
12
u4u3
u2u1
u=100
u=100 u=100
u=0L = 12 cm , h= 4 cmN= 12/4 = 3
-4uij+ uE+uW+uN+uS = 0
(1) -4u1+u2+100+100+u3 = 0
(2) -4u2+u1+100+100+u4 = 0
(3) -4u3+100+u4+u1+0 = 0
(4) -4u4+100+u3+u2+0 = 0
SE301_Topic 6 Al-Amer2005 6
In matrix form Au=b
100
100
200
200
4110
1401
1041
0114
4
3
2
1
u
u
u
u
But the problem is symmetry, u1 =u2 and u3 =u4
So we can solve only for u1 and u3
(1) -4u1+u2+100+100+u3 = 0 (3) -4u3+100+u4+u1+0 = 0
Then replace for u1 =u2 and u3 =u4, the system of equations is reduced to
-3u1 +u3 = -200
u1-3u3 = -100
Which has a solution u1 =u2 =87.5 and u3 =u4 =62.5
Example 2Solve the mixed BVP for the poisson equation (uxx + uyy =12xy) on a rectangle plate as shown in the fig. (take x =y =0.5)
x = 0 0.5 1 1.5
0
0
.5
1
u4u3
u2u1
u=0
u=0 u=3y3
xdy
du6
u=0.375
u=3
xyy
u
x
u12
2
2
2
2
solution
x = 0 0.5 1 1.5 0
0.5
1
u4u3
u2u1
u=0
u=0u=3y3
-4uij+ uE+uW+uN+uS = 12h2xy=3xy
(1) -4u1+u2+u3 =3(0.5)(0.5)
(2) -4u2+0.375+u1+u4 = 3(1)(0.5)
(3) -4u3+u4+u1+u5 = 3(0.5)(1)
(4) -4u4+3+u3+u2+u6 = 3(1)(1)
xdy
du6
u6u5
u=0.375
u=3
x = 0 0.5 1 1.5 0
0.5
1
u4u3
u2u1
u=0
u=0u=3y3
xdy
du6
u6u5
u=0.375
u=3
xdy
du6
3)5.0(6)5.0(2 1515
uu
uu
6)1(6)5.0(2 2626
uu
uu
Then replace for u5 and u6
And put the system in the matrix form
SE301_Topic 6 Al-Amer2005 10
6
5.1
125.1
75.0
4120
1402
1041
0114
4
3
2
1
u
u
u
u
Tu 191.0077.0812.1866.0
Which has a solution
EXERCISE
For a rectangle thin plate of dimension 4*3 units, u(x,0)=u(x,3)=10x, u(0,y)=0 and u(4,y)= 40 +10y(y-3). Solve the Poisson’s equation 2u = 5x , take x = y = 1
EXERCISES