  We combine the newly-constructed Galerkin difference basis with the
energy-based discontinuous Galerkin method for wave equations in second order
form. The approximation properties of the resulting method are excellent and
the allowable time steps are large compared to traditional discontinuous
Galerkin methods. The one drawback of the combined approach is the cost of
inversion of the local mass matrix. We demonstrate that for constant
coefficient problems on Cartesian meshes this bottleneck can be removed by the
use of a modified Galerkin difference basis. For variable coefficients or
non-Cartesian meshes this technique is not possible and we instead use the
preconditioned conjugate gradient method to iteratively invert the mass
matrices. With a careful choice of preconditioner we can demonstrate optimal
complexity, albeit with a larger constant.
