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## Section: New Results

### Time integration strategies and resolution algorithms

#### Hybrid explicit-implicit DGTD-${ℙ}_{p}$ method

Participants : Stéphane Descombes, Stéphane Lanteri, Ludovic Moya.

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 meshes. An implicit time integration scheme is a natural way to obtain a time domain method which is unconditionally stable. Starting from the explicit, non-dissipative, DGTD-${ℙ}_{p}$ method introduced in [14] , we have proposed the use of a Crank-Nicolson scheme in place of the explicit leap-frog scheme adopted in this method [4] . As a result, we obtain an unconditionally stable, non-dissipative, implicit DGTD-${ℙ}_{p}$ method, but at the expense of the inversion of a global linear system at each time step, thus obliterating one of the attractive features of discontinuous Galerkin formulations. A more viable approach for 3D simulations consists in applying an implicit time integration scheme locally i.e. in the refined regions of the mesh, while preserving an explicit time scheme in the complementary part, resulting in an hybrid explicit-implicit (or locally implicit) time integration strategy. Such an approach, combining a leap-frog scheme and a Crank-Nicolson scheme, has been studied numerically in [6] , showing promising results which have motivated further investigations on theoretical issues (especially, convergence in the ODE and PDE senses) [28] .

#### Explicit local time stepping DGTD-${ℙ}_{p}$ method

Participants : Joseph Charles, Julien Diaz [MAGIQUE-3D project-team, INRIA Bordeaux - Sud-Ouest] , Stéphane Descombes, Stéphane Lanteri.

We have initiated this year a collaboration with the MAGIQUE-3D project-team aiming at the design of local time stepping strategies inspired from [41] for the time integration of the system of ordinary differential equations resulting from the discretization of the time domain Maxwell equations in first order form by a DGTD-${ℙ}_{p}$ method. A numerical study in one- and two-space dimensions is underway.

#### Optimized Schwarz algorithms for the frequency domain Maxwell equations

Participants : Victorita Dolean, Mohamed El Bouajaji, Martin Gander [Mathematics Section, University of Geneva] , Stéphane Lanteri, Ronan Perrussel [Laplace Laboratory, INP/ENSEEIHT/UPS, Toulouse] .

We continued with the design of optimized Schwarz algorithms for the solution of the frequency domain Maxwell equations. In particular, we have analyzed a family of methods adapted to the case of conductive media [21] . Besides, we have also proposed discrete variants of these algorithms in the framework of a high order discontinuous Galerkin discretization method formulated on unstructured triangular meshes for teh siolution of the 2D time harmonic Maxwell equations.