Finite-Difference (FD) Solutions to Elliptic PDE

1
ME6758 Dr. Ferri
Finite-Difference (FD) Solutions to Elliptic PDEs
y
h(b-a)/n
d
k(d-c)/m
c
x
b
a
2
Example
a0, b0.5, c0, d0.5, m4, n4
u200x
P1
P2
P3
(3,3)
(2,3)
(1,3)
P4
P5
P6
u0
u200y
(1,2)
(2,2)
(3,2)
P7
P8
P9
(1,1)
(2,1)
(3,1)
wL w1,j
L i (m 1 j )(n-1)
u0
3
Example
a0, b0.5, c0, d0.5, m4, n4
A 4 -1 0 -1 0 0 0
0 0 -1 4 -1 0 -1 0
0 0 0 0 -1 4 0 0
-1 0 0 0 -1 0 0 4
-1 0 -1 0 0 0 -1 0
-1 4 -1 0 -1 0 0 0
-1 0 -1 4 0 0 -1 0
0 0 -1 0 0 4 -1 0
0 0 0 0 -1 0 -1 4
-1 0 0 0 0 0 -1 0
-1 4
B 25 50 150 0 0 50
0 0 25
4
Example
a0, b0.5, c0, d0.5, m4, n4
5
Same example, finer mesh
a0, b0.5, c0, d0.5, m20, n20
6
Different Dirichlet BCs
a0, b0.5, c0, d0.5, m20, n20
u50sin(py/0.5)
u0, three sides
7
Example with nonzero f(x,y)
Heat source
a0, b0.5, c0, d0.5, m20, n20
Zero temp, all sides
8
Example
u200x
u0
a0, b0.5, c0, d0.5, m4, n4
u200y
insulated
9
Insulated Lower Edge
No change
A 4 -1 0 -1 0 0 0
0 0 0 0 0 -1 4 -1
0 -1 0 0 0 0 0 0
0 0 -1 4 0 0 -1 0
0 0 0 0 0 -1 0 0
4 -1 0 -1 0 0 0 0
0 0 -1 0 -1 4 -1 0
-1 0 0 0 0 0 0 -1
0 -1 4 0 0 -1 0 0
0 0 0 0 -1 0 0 4
-1 0 -1 0 0 0 0 0
0 -1 0 -1 4 -1 0 -1
0 0 0 0 0 0 -1 0
-1 4 0 0 -1 0 0 0
0 0 0 -2 0 0 4 -1
0 0 0 0 0 0 0 0
-2 0 -1 4 -1 0 0 0
0 0 0 0 0 -2 0 -1 4
B 25 50 150 0 0 50
0 0 25 0 0 0
No change
Shows the effect of insulated BC