Also, your equations look wrong to me. It seems like you are trying to model a scalar wave equation \ddot{x} = \nabla^2 x, but your equations are not quite a discretization of this equation. For example, the first term in the first equation should be multiplied by twice the number of dimensions.
In general, I would look into discretization schemes for hyperbolic equations (e.g. the leapfrog scheme I mentioned), because there are lots of ways to go wrong here. Or you could use DifferentialEquations.jl to handle the time integration (i.e. “method of lines”).
Note that a useful technique is to use “ghost cells” to handle boundary conditions, as described here: Finite difference Laplacian with five-point stencil - #2 by stevengj