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2.4.1 Explicit Time Integration

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