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This study is concerned with the solution of the time domain Maxwell's equations in a dispersive propagation media by a Discontinuous Galerkin Time Domain (DGTD) method. The Debye model is used to describe the dispersive behaviour of the media. The resulting system of equations is solved using a centered flux discontinuous Galerkin formulation for the discretization in space and a second order leapfrog scheme for the integration in time. The numerical treatment of the dispersive model relies on an Auxiliary Differential Equation (ADE) approach similary to what is adopted in the Finite Difference Time Domain (FDTD) method. Stability estimates are derived through energy estimations and the convergence is proved for both the semi-discrete and the fully discrete case.

International audienceWe investigate numerically the two dimensional travelling waves of the Nonlinear Schrödinger Equation for a general nonlinearity and with nonzero condition at infinity. In particular, we are interested in the energy-momentum diagrams. We propose a numerical strategy based on the variational structure of the equation. The key point is to characterize the saddle points of the action as minimizers of another functional, that allows us to use a gradient flow. We combine this approach with a continuation method in speed in order to obtain the full range of velocities. Through various examples, we show that even though the nonlinearity has the same behaviour as the well-known Gross-Pitaevskii nonlinearity, the qualitative properties of the travelling waves may be extremely different. For instance, we observe cusps, a modified (KP-I) asymptotic in the transonic limit, various multiplicity results and ''one dimensional spreading'' phenomena

Explicit solitary waves are known to exist for the Kadomtsev-Petviashvili-I (KP-I) equation in dimension 2. We first address numerically the question of their Morse index. The results confirm that the lump solitary wave has Morse index one and that the other explicit solutions correspond to excited states. We then turn to the 2D Gross-Pitaevskii (GP) equation, which in some long wave regime converges to the KP-I equation. Numerical simulations have already shown that a branch of travelling waves of GP converges to a ground state of KP-I, expected to be the lump. In this work, we perform numerical simulations showing that other explicit solitary waves solutions to the KP-I equation give rise to new branches of travelling waves of GP corresponding to excited states.

Abstract. In this paper, we are concerned with the numerical modelling of the propagation of electromagnetic waves in dispersive materials for nanophotonics applications. We focus on a generalized model that allows for the description of a wide range of dispersive media. The underlying differential equations are recast into a generic form and we establish an existence and uniqueness result. We then turn to the numerical treatment and propose an appropriate Discontinuous Galerkin Time Domain framework. We obtain the semi-discrete convergence and prove the stability (and in a larger extent, convergence) of a Runge Kutta 4 fully discrete scheme via a technique relying on energy principles. Finally, we validate our approach through two significant nanophotonics test cases.

Abstract.As an example of a simple constrained geometric non-linear wave equation, we study a numerical approximation of the Maxwell Klein Gordon equation. We consider an existing constraint preserving semi-discrete scheme based on finite elements and prove its convergence in space dimension 2 for initial data of finite energy.Mathematics Subject Classification. 65M60, 78M10.

International audienceThe present work is about the development of a parallel non-conforming multi-element discontinuous Galerkin time-domain (DGTD) method for the simulation of the scattering of electromagnetic waves by metallic nanoparticles. Such nanoparticles most often have curvilinear shapes, therefore we propose a numerical modeling strategy which combines the use of an unstructured tetrahedral mesh for the discretization of the scattering structures with a structured (uniform cartesian) mesh for treating efficiently the rest of the domain. The overall goal is to increase the flexibility in the meshing process while decreasing the needs in computational resources for the target applications. The latter are here modeled by the system of 3D time-domain Maxwell equations coupled to a Drude dispersion model for taking into account the material properties of nanoparticles at optical frequencies. We propose an auxiliary differential equation (ADE) based DGTD method for solving the resulting system and present numerical results demonstrating the benefits of using non-conforming multi-element meshes in this particular application context

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