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solving an elliptic equation. The first is that solution is required simultaneously for each
grid. This means that for a large computational domain, say m by n grids, it is necessary
to invert anmxnbymxn matrix. If m and n are sufficiently large, problems may be
encountered in terms of the interned memory storage, and even if the available storage is
not a problem, the number of computations required to invert the matrix may make for
an exceedingly slow solution. In some situations this may be acceptable; however for the
purposes of a wave induced circulation model in which repeated solution for the wave field
is necessary the program would take a very long time to run. The other disadvantage in
volved in the numerical solution of an elliptic equation is that boundary conditions must be
specified at all of the boundaries. In many applications the only known boundary condition
is the offshore, or incident wave amplitude. For example, the domain of interest may be the
wave field in the lee of an obstruction where the only down wave condition is that the waves
freely pass through the down wave boundary. Or in the present study in which the wave
amplitude is known to go to zero at the shoreline a problem ensues in that the shoreline is
(for purposes of the circulation model) at the grid edge and not at the grid point.
A parabolic equation eliminates both of these problems. Since the solution proceeds grid
row by grid row where each solution uses the results of the previous grid row, no down wave
information is required. Since only one grid row is solved at a time, the solution requires
only that a tridiagonal matrix equation be solved to obtain values for the grid row. The only
required boundary conditions are the conditions on the first grid row (usually the offshore
boundary) and lateral boundary conditions which are usually a derivative condition that
allows for the wave to pass through the boundary without any reflection at the boundary.
There are several ways to transform the elliptic Eq. (3.37) to a parabolic equation. The
most direct is to make certain assumptions concerning the length scale of variations in the x
direction. At the heart of the parabolic approximation is the assumption that the direction
of wave propagation is essentially along the x axis. For waves propagating at an angle to