Lecture Notes
ME4906, Mechanical Engineering, PolyU
2008/2009
5.3 Finite Difference for PDE of 1D in Space
5.3.1 Explicit Scheme
Consider the problem of 1D unsteady heat transfer defined below,
The solution domain is two-dimensional, space
x
and time
t
. The length L is divided
into
N
x
points and the time advances from
t=
0 at an interval of
∆
t
:
t
=(1,2,3,…)
t
.
Figure 4. Explicit scheme for time derivative
The second space derivative is approximated by
()
( ) ( ) ( )
2
1,
2
,
,
−−
++
=∆ =∆ ≈
∆
xx
Ti
l
Til Ti
l
Txi
x
tl
t
x
and the time derivative is given by
( ) ( )
,1
,
,
+−
∆
t
Til
Tx ix
t lt
t
So the PDE is discretized as
(
)
( ) ( )
(
)
2
2
,
1
, 2 ,
1
,
,1 12
,
1
,
1
, ,
/
κ
− −
+ +
=
∆∆
∴
+=−
+
− + +
=∆∆
T il
T i
l
l
tx
cTil cTi
l Ti
l
c
t x
Time
x=L
x
t
i-
1,
i,
i+
1
l+
1,
l
2
2
∂∂
=
TT