CHAPTER 4
FINITE DIFFERENCE SCHEMES AND BOUNDARY CONDITIONS
4.1 Numerical Solution of the Governing Equations
The finite differencing of the governing equations in the circulation model is derived
by a matrix analysis as suggested by Sheng and Butler (1982). A rational analysis of the
complete Eqs. (2.1-2.3) including all the nonlinear terms is not possible. However guidance
can be obtained through an analysis of the linearized equations. For this purpose the linear
equations for the special case of a flat bottom and without any of the forcing terms will be
examined.
dfj au av n
-7T + D + D = 0
dt dx dy
au dfj
lt+3dz=
d-X. + gdA = 0
at ydy
Let W be a vector of fj,U and V
W =
U
V
Using Eq. (4.4), the governing equations are written as
(4.1)
(4.2)
(4.3)
(4.4)
Wt + AWX + BWy = 0
where A and B are the following matrices
(4.5)
' 0
D
0 '
' 0
0
D '
A =
9
0
0
and
B =
0
0
0
0
0
0
. 9
0
0
(4.6)
To find an implicit finite difference scheme for the governing equations, Eq. (4.5) is
finite differenced as
jyn+i jyn
AT
+ AW?+1 + BW^1 = 0
(4.7)
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