Taking the forward time central space technique [15] into (5), we get the following

finite difference equation:

The

finite difference equation of the second term, [K.sub.[theta]][partial derivative][h.sup.*]/[partial derivative]z in Eq.

The

finite difference equation for the difference momentum equation can be written as

Substituting (7), (10), and (13) into (14) and replacing [u.sub.m+i,n+j] with [U.sub.m+i,n+j] (i, j [member of] [Z.sub.2]) yield the 25-point

finite difference equationAlthough the use of a uniform grid system, like that shown in Figure 2, greatly simplifies the

finite difference equations and cell-to-cell relationships, it is computationally inefficient.

In fact the essence of the method of differences for the solution of differential equations is that instead of solving a differential equation one solves a corresponding

finite difference equation that is obtained by substituting differences expressions with higher or lower level of accuracy for the derivatives.

Finite difference inequalities which exhibit explicit bounds on unknown functions in general provide a very useful and important tool in the development of the theory of

finite difference equations. Due to the fact that finite difference inequalities have been applied in various branches of

finite difference equations, many such inequalities with in one or two variables have been established [1-4].

Richtmyer, Survey of the stability of linear

finite difference equations, Comm.

Richtmyer, "Survey of the stability of linear

finite difference equations," Communications on Pure and Applied Mathematics, vol.

One idea is to replace the derivatives appearing in the partial differential equation by

finite difference equations that approximate them.

Centered differencing at point (i' k) is achieved on these equations, leading to

finite difference equations