We note that the system of equations (2)--(4) has both parabolic and hyperbolic properties, the former associated with the diffusion terms and the latter with the pressure gradients and nonlinear terms. Diffusion of momentum will lead (by itself) to a parabolic partial differential equation. The coupling of the time derivatives (in all three equations) to the pressure gradients and the Coriolis terms (in equations (2) and (3)) will lead to a system that describes inertia-gravity waves and has hyperbolic characteristics. (Note: inertia-gravity waves here refer to gravity waves where the restoring buoyancy force is modified by the rotation of the Earth by the Coriolis force).

The
basic techniques for the numerical solutions to these various
classes of partial differential equations are given in the PDE
chapter of this book. For geophysical fluid dynamic problems, an
extensive survey of more specific finite-difference schemes has
been given by Grammeltvedt
[24]
and Arakawa and Mesinger
[45].
Another collection of numerical methods in oceanography can be
found in O'Brien
[51].
As the reader will see from both the
outside literature and the various chapters of this book, the
numerical schemes for solving time-dependent partial differential
equations fall generally into two classes: explicit or implicit.
To explain this terminology, let us assume that all variables are
evaluated at discrete time and space intervals such that
represents and is approximated as
.
Then the term ``explicit'' denotes a scheme where all
terms on the right hand sides (r.h.s.) of system
(2)--(4)
are
evaluated at time steps **n**, **n-1**, etc., i.e. at any given time the
r.h.s. is known from previous time steps. On the other hand,
``implicit'' denotes a scheme where some of the terms on the r.h.s.\
are evaluated at time step and thus are not known at time .
In order to proceed then, one needs to transfer these terms to the
left hand sides (l.h.s) of the equations and invert the
corresponding coefficient matrix for the unknown variables at
.