Page 101 - Standard Handbook Of Petroleum & Natural Gas Engineering

90 Mathematics

with boundary conditions

u(a,y) = ua
uby) = u,,

U(X,O) = u,,
the equation can be written in a finite difference form, considered over the grid
as shown in Figure 1-58. Using a central difference form for the derivative with
respect to x and a forward difference form for the derivative with respect to
y gives

and

If j is set to zero, the procedure can be used to take the first step after the
initial conditions are set.
If azu/ax2 is represented by a central difference expression and h/ay by a
backward difference expression an implicit solution may be obtained where

V

X
1
Figure 1-58. Finite difference grid.