We introduce a new numerical method for the time-dependent Maxwell equations
on unstructured meshes in two space dimensions. This relies on the introduction
of a new mesh, which is the barycentric-dual cellular complex of the starting
simplicial mesh, and on approximating two unknown fields with integral
quantities on geometric entities of the two dual complexes. A careful choice of
basis-functions yields cheaply invertible block-diagonal system matrices for
the discrete time-stepping scheme. The main novelty of the present contribution
lies in incorporating arbitrary polynomial degree in the approximating
functional spaces, defined through a new reference cell. The presented method,
albeit a kind of Discontinuous Galerkin approach, requires neither the
introduction of user-tuned penalty parameters for the tangential jump of the
fields, nor numerical dissipation to achieve stability. In fact an exact
electromagnetic energy conservation law for the semi-discrete scheme is proved
and it is shown on several numerical tests that the resulting algorithm
provides spurious-free solutions with the expected order of convergence.Comment: 34 pages, 14 figures, submitte