Numerically Approximating the Wave Equation?
Take a look at what is known as "symplectic integration" and also "discrete exterior calculus" (DEC). Both of these should work, and they use some heavy math to justify the methods. Its quite entertaining. The basic outline for DEC is that, in vacuum, you have four constraint equations which much be satisfied (there's not any stray electric or magnetic fields) and so the equations evolve in such a way as to strictly keep this at zero. The algebra of the constraints dictates where answers can be found from one timestep to the next, so it improves stability to use a calculus which makes sure that the algebraic properties of the constraints are maintained. I hope that helps. Feel free to email me.