Contents I. Introduction
To evaluate the fields in electromagnetic problems, the finite-difference time-domain (FDTD) method is a well-known method to solve Maxwell's equations in the time domain [1], [2]. However, FDTD presents some problems such as the staircase approach of the shape of the objects [3] and the numerical dispersion phenomenon. To improve on these points, many authors have studied alternative methods, for example based on finite-volume time-domain (FVTD) or finite integration technique (FIT) [4], [5] or the finite-element time-domain (FETD) formalism. In the FVTD methods, the Maxwell equations are written in conservative form. The fields, located generally at the center of a cell in the mesh, are evaluated as the sum of fluxes taken at the faces of the cell. Depending on the choice of the flux, FVTD techniques lead to either stable but dissipative methods [6]–[9], methods with weak instabilities for non orthogonal grids [10]– [12] or, more recently, stable and nondissipative but more dispersive methods [13]. To improve these methods, several authors introduce a high spatial order approximation for the solution in each cell. Such numerical schemes are based on a discontinuous Galerkin formulation [14]. Despite their great accuracy, and the fact that they have block-diagonal mass-matrices, these schemes suffer from an a large number of unknowns and, hence, an over-cost in terms of computation. Indeed, the discontinuity of the solution, which these schemse allow on all the cell faces, need to duplicate systematically the unknowns located at the boundaries of the cells, even when the actual field is continuous.
