Page 131 - Computational Fluid Dynamics for Engineers

4.5 Finite-Difference Methods for Elliptic Equations 117

2
uij - 0 x(u 2j + u 0j) - 0 y(uij+i + uij-i) = S fij = Fij
and, with Eq. (4.5.15a), it can be written as
4 1
uij - 0 x(u 2j + -uij - -u 2j) - 0 y(u!j+i + uij-x) = Fij
or as

u
0
u
1
V ~ 3 ^ ) Ul ^~3 0xU2 d~ y( hj+i^ hj-i) = F hj> 2 <j< J- 1 (4.5.16a)
For i — J, Eq. (4.5.4b) becomes
u
v>l,j ~ Ox(u>i+ij + i-i,j) ~ Oy(u>ij+i + u j j - i ) = F / j
and with the boundary condition at i = I + 1, that is,

UI+IJ = 0 (4.5.15c)
Eq. (4.5.4b) at i = I can be written as

UIJ - O XUI-IJ - O y(uij+i + UIJ-I) = F / j 2 < j < J - 1 (4.5.16b)

For j = 1, Eq. (4.5.4b) becomes
^z,i - Oxiui+1,1 + Wi-i,i) - 0 y(ui2 + tt;,o) = Fi,i

and, with Eq. (4.5.15b), can be written as

4 1
Ui,i - O x(u i+i,i + Ui-i ti) - 0 y(ui2 + o^i,i - 0 ^ , 2 ) = ^ , 1
1
o ^ ' " 3
or as
1
e u
Ui 1
y
1
x Ui
1
\ ~ 3° ) Ui > ~° ( + > + - ^~% y i2 = ^ . i ' 2<i< 1-1 (4.5.16c)
For j = J, Eq. (4.5.4b) becomes
Ui,j - O x(u i+1j + Ui_^j) - e y(u ijJ+i + Ui,j_i) = F^j
and with the boundary condition at j = J + 1, that is,
Ui,j+i = 0 (4.5.15d)

Eq. (4.5.4b) at j = J can be written as

^i,j - Ox{ui+i,j + ^ - i , j ) - O y(uij-i) = F i ; J , 2 < i < J - 1 (4.5.16d)
At i = j = 1, Eq. (4.5.4b), with the relations given by Eqs. (4.5.15a,b), becomes

/ 4 4 \ 2 2
( 1 - g#x - -6 y \ ^1,1 - -0 xu 2,\ - -0 yui,2 = Fi,i (4.5.17a)