7.5 Two-Dimensional Problems 421
Figure 6 Mesh numbering for numerical solution of Eqs. (2)–(5).We t a k e x= y= 1 /4 and number the interior points of the region as
shown in Fig. 6. Then we will be computing the approximations
u 1 (m)∼=u( 1
4
,^1
4
,tm)
, u 2 (m)∼=u( 1
2
,^1
4
,tm)
, u 3 (m)∼=u( 3
4
,^1
4
,tm)
,...
(6)
and so forth, form= 1 , 2 ,.... The replacement equations are obtained using
Eq. (1) for the Laplacian and a forward difference to replace the time deriva-
tive. The typical equation is
uN(m)+uS(m)+uE(m)+uW(m)− 4 ui(m)
( x)^2=ui(m+^1 )−ui(m)
t. (7)
Whenwesolvethisequationforui(m+ 1 ),weobtain
ui(m+ 1 )=r[uN(m)+uS(m)+uE(m)+uW(m)]+( 1 − 4 r)ui(m), (8)in which
r=t
x^2 =t
y^2 =^16 t.The stability considerations of Section7.2 are still important, and the rules of
thumb are still valid. We must limitrby the requirement that 1− 4 r≥0, or, in
this case, t≤ 1 /64. We shall take the longest acceptable time step, t= 1 /64,
r= 1 /4, which makes the equations a little simpler.
Atm=0,alltemperaturesaregivenas1.Form≥1, all the boundary tem-
peratures are zero and theui(m)are all found to equal 1. Form=2, we calcu-
late
u 1 ( 2 )=1
4
(
u 2 ( 1 )+u 5 ( 1 )+ 0 + 0)
=
1
2 ,
u 2 ( 2 )=^1
4(
u 1 ( 1 )+u 3 ( 1 )+u 6 ( 1 )+ 0