Our model problem is the 2D wave equation,
with fixed boundaries (
and 
at the boundaries).
We split the second order equations into a system of 3 first order partial
differential equations,
where we have defined
To solve the first order set of equations (3) we use a non-staggered leapfrog technique where we difference using a stencil like the one shown in figure 1.
The difference equations used to solve the wave equation are as follows:
