52
is finite differenced as pF\Un\Un+1 and the friction term as suggested by Longuet-Higgins is
differenced as pF |uorj| Un+1, where |uorj| is in terms of values at the n time level. However,
when using the integral quadratic expression, it is evaluated entirely in terms of the present
or known values.
The nonlinear advective acceleration terms are added at the ntfl time level using central
difference derivatives. In the x direction momentum equation these terms are
7T + ^ilj i 1 [ir i 1/ i t/- i -tr \ ^J+l 1
u,j 2AX h + v*-iJ + vJ+l + ^y
(4.24)
and in the y direction equation they are
jWj + U<-U + ViJ+l + (4.25)
In the model of Vemulakonda (1984) which is based upon the WIFM (Waterways Ex
periment Station Implicit Flooding Model) (Butler 1978) model, a three time level leap-frog
scheme is used. This gives the advantage that all of the forcing terms (and the nonlinear
acceleration terms) are added at the time centered level. However this has some disad
vantages in that additional computer memory and computational time are required and
leap-frog schemes can exhibit instabilities. In the present study a two time level scheme
is used and the wave forcing terms as well as the non-linear acceleration terms are added
at the present time level. Since, for all of the applications in this report, the model is run
until steady state conditions are obtained, it is reasoned that there is no great necessity to
properly time center all of the terms. If the use of the model is extended to time dependent
applications, this may need to be corrected.
4.2 Finite Differencing of the Parabolic Wave Equation
Dividing Eq. (3.82) by (Cycosi + U) yields
f C, cos B+U j
A'x + i(k cos k cos 6) A1 + ^ ^
;A' +
+
2 (C,cos+ 17) (C,cos + 7) v
, i kCa(l cosJ ) A,
v 2 (Ct cos + U)
(Y.)
2(Ccos + 7) \a J
2(Ce cos + U)
CC,
+
w
2(Cy cos 0 + U)
A' = 0
(4.26)