A high order multilevel numerical solver for Maxwell's equations using local discrete convolutions is presented. Maxwell's equations are embedded in a system of wave equations. The solution to this system can be derived with the method of spherical means. When the source terms are prescribed functions then there is an explicit update formula using the propagator form of the solution and Duhamel's formula. And when the source terms are dependent on the fields, the propagator can be embedded in the system of differential equations by a change of variables by Lawson's method. The spherical means operations are discretized using the framework from Tornberg and on uniform spacing rectangular grids the procedure becomes constant coefficient stencil operations. While the source term integration is handled with a quadrature scheme or a time integrator. This method can be parallelized with standard domain decomposition. Since this method places the electric and magnetic fields on the same grids, as opposed to staggered grids used in standard grid based solvers for Maxwell's equations, and that it has no time stepping restriction, because it is a propagator based method, it can be extended to use local mesh refinement with only simple interpolating and sampling operators.