Recent advances in manufacturing techniques, such as electron beam lithography, have made possible the production of electromagnetic devices on the scale of the wavelength of light. Resonant cavities on this scale have high quality factors and well defined resonant frequencies and are accurately described by Maxwells' equation. A constitutive model describing polarisation, based on the Drude model of solids, is also required to model media which exhibit significant frequency dispersion. Traditionally, quantities of interest, such as resonant frequencies, quality factors and the associated mode shapes, are calculated using a frequency domain approach. However, computations over broad frequency ranges require numerous simulation runs and, for more geometrically complex problems, solving the resulting large eigenvalue system quickly becomes computationally prohibitive.

We propose a time-domain approach, using a high-order, parallelised Discontinuous Galerkin (DG) solver with explicit time marching. The broad band frequency response of the system is then obtained by periodically sampling a signal from the solution of a single time-domain simulation and applying a frequency domain transform, such as the fast Fourier transform. We will discuss the need for high sampling rates and long-running simulations in order to obtain broad, high-resolution frequency spectra. The implementation will be validated both in 2D and 3D, by comparing to problems with known analytical solution. A detailed study of the optimal convergence properties of the method will be presented, including quantifying the effect of the geometric approximation of curved boundaries on the convergence rate. The inclusion of Drude model polarisation will be validated and illustrated using examples. The implementation of a parallel 3D DG solver will be discussed, and mesh partition optimisation strategies will be presented for hybrid meshes and high-order meshes with variable order elements. Finally we will demonstrate the ability of the method to obtain accurately resolved spectra for realistic examples of 3D cavities in a reasonable computational time.