Geology Reference
In-Depth Information
Finite difference solution for partial differential equation: Anything moving
on earth is governed by partial differential equation. To solve this partial
differential equation we discretize the area into equal length area. There are
two methods for solving this partial differential equations: the Explicit method
and the Implicit method.
Here we will take the case of 1-D unsteady groundwater flow equation.
It could be written as,
2
--
u
u
(17)
2
-
t
-
x
3$ 0 x $' 1, t 0
Figure 4. A grid for a finite difference solution.
Initial condition u ( x , 0) = u 0
1st Boundary condition u (0, t ) = u 0 , 3 t > 0
2nd Boundary condition u (1, t ) = u 1 ,
t > 0
Now we consider a function f ( a ), which is smooth. Then, f may be expanded
into Taylor series about a in the positive direction
3
h
2
fa
+ -----------
(18)
f ( a + h )= f ( a ) + hf
( a ) +
2!
h
h
2
fa h
(
h = f ( a ) +
)
fa
( )
f
a
fa
+
+ ---------
2!
3!
if h << 0 then,
 
Search WWH ::




Custom Search