Une méthode volume fini implicite en maillages non-structurés pour les équations de Maxwell 3D en domaine temporel

Existing numerical methods for the solution of the time domain Maxwell equations often rely on explicit time integration schemes and are therefore constrained by a stability condition that can be very restrictive on highly refined or unstructured meshes. The present study aims at investigating the applicability of an implicit time integration scheme in conjunction with a finite volume approximation method on unstructured meshes. If one wants to achieve at least second-order accuracy in time, then the most natural choice for the implicit time integration of the semi-discrete Maxwell equations is the Crank-Nicolson scheme. In the first part of the paper, we study some of the mathematical properties of the resulting Implicit Finite Volume Time Domain (IFVTD) method in the one dimensional case using different boundary conditions (metallic, periodic and absorbing boundary conditions). We prove that the IFVTD method globally conserves a discrete form of the electromagnetic energy if the initial boundary value problem is based on metallic or periodic boundary conditions. If an absorbing boundary condition is used then we show that the discrete electromagnetic energy is decreasing. In this one dimensional study, we also prove that the matrix operator characterizing the IFVTD method is invertible. In the second part of the paper, we extend the results of the one dimensional analysis to the three dimensional Maxwell equations. We conclude with preliminary numerical results on a simple academic test case in order to validate the proposed method