We commence with explicit time marching schemes
because they do not involve matrix inversion or iterative solvers.
Before we write down the complete finite difference analog of
equations
(2)--(4),
we must give finite difference expressions to
the first and second spatial derivatives in the **x**- and
**y**-directions. We will use only centered differences unless otherwise
stated. Denoting , we have

Though we avoid having to use a matrix inverter to solve for the values at , we pay the price by being restricted in the time step we can take. The size of a time step one can take in solving time-marching differential equations by explicit methods is governed by the famous Courant--Friedrichs--Levy (CFL) stability condition. For wave equations the time step is limited by the wave speed, in this case the speed of the surface gravity waves , and is given by