  Maxwell's equations are considered with transparent boundary conditions, for
initial conditions and inhomogeneity having support in a bounded, not
necessarily convex three-dimensional domain or in a collection of such domains.
The numerical method only involves the interior domain and its boundary. The
transparent boundary conditions are imposed via a time-dependent boundary
integral operator that is shown to satisfy a coercivity property. The stability
of the numerical method relies on this coercivity and on an anti-symmetric
structure of the discretized equations that is inherited from a weak
first-order formulation of the continuous equations. The method proposed here
uses a discontinuous Galerkin method and the leapfrog scheme in the interior
and is coupled to boundary elements and convolution quadrature on the boundary.
The method is explicit in the interior and implicit on the boundary. Stability
and convergence of the spatial semidiscretization are proven, and with a
computationally simple stabilization term, this is also shown for the full
discretization.
