  In this paper, we consider Maxwell's equations in linear dispersive media
described by a single-pole Lorentz model for electronic polarization. We study
two classes of commonly used spatial discretizations: finite difference methods
(FD) with arbitrary even order accuracy in space and high spatial order
discontinuous Galerkin (DG) finite element methods. Both types of spatial
discretizations are coupled with second order semi-implicit leap-frog and
implicit trapezoidal temporal schemes studied in our previous research [5,6].
By performing detailed dispersion analysis for the semi-discrete and fully
discrete schemes, we obtain rigorous quantification of the dispersion error for
Lorentz dispersive dielectrics. In particular, comparisons of dispersion error
can be made taking into account the model parameters, and mesh sizes in the
design of the two types of schemes. The results for the numerical dispersion
analysis can guide us in the optimal choice of discretization parameters for
the more complicated and nonlinear models. The numerical dispersion analysis of
the fully discrete FD and DG schemes, for the dispersive Maxwell model
considered in this paper, clearly indicate the dependence of the numerical
dispersion errors on spatial and temporal discretizations, their order of
accuracy, mesh discretization parameters and model parameters. The results
obtained here cannot be arrived at by considering discretizations of Maxwell's
equations in free space. In particular, our results contrast the advantages and
disadvantages of using high order FD or DG schemes and leap-frog or trapezoidal
time integrators over different frequency ranges using a variety of measures of
numerical dispersion errors. Finally, we highlight the limitations of the
second order accurate temporal discretizations considered.
