Hybrid resonance and long-time asymptotic of the solution to Maxwell's equations

We study the long-time asymptotic of the solutions to Maxwell's equation in the case of a upper-hybrid resonance in the cold plasma model. We base our analysis in the transfer to the time domain of the recent results of B. Despr\'es, L.M. Imbert-G\'erard and R. Weder, J. Math. Pures Appl. {\bf 101} ( 2014) 623-659, where the singular solutions to Maxwell's equations in the frequency domain were constructed by means of a limiting absorption principle and a formula for the heating of the plasma in the limit of vanishing collision frequency was obtained. Currently there is considerable interest in these problems, in particular, because upper-hybrid resonances are a possible scenario for the heating of plasmas, and since they can be a model for the diagnostics involving wave scattering in plasmas.Comment: This published version has been edited to improve the presentation of the result

Computation of sum of squares polynomials from data points

We propose an iterative algorithm for the numerical computation of sums of squares of polynomials approximating given data at prescribed interpolation points. The method is based on the definition of a convex functional $G$ arising from the dualization of a quadratic regression over the Cholesky factors of the sum of squares decomposition. In order to justify the construction, the domain of $G$, the boundary of the domain and the behavior at infinity are analyzed in details. When the data interpolate a positive univariate polynomial, we show that in the context of the Lukacs sum of squares representation, $G$ is coercive and strictly convex which yields a unique critical point and a corresponding decomposition in sum of squares. For multivariate polynomials which admit a decomposition in sum of squares and up to a small perturbation of size $\varepsilon$, $G^\varepsilon$ is always coercive and so it minimum yields an approximate decomposition in sum of squares. Various unconstrained descent algorithms are proposed to minimize $G$. Numerical examples are provided, for univariate and bivariate polynomials

Finite Volume Transport Schemes

We analyze arbitrary order linear nite volume transport schemes and show asymptotic stability in L1 and L1 for odd order schemes in dimen- sion one. It gives sharp fractional order estimates of convergence for BV solutions. It shows odd order nite volume advection schemes are better than even order nite volume schemes. Therefore the Gibbs phenomena is controled for odd order nite volume schemes. Numerical experiments sustain the theoretical analysis. In particular the oscillations of the Lax- Wendro scheme for small Courant numbers are correlated with its non stability in L1. A scheme of order three is proved to be stable in L1 and L1

Non linear finite volume schemes for the heat equation in 1D.

We construct various explicit non linear finite volume schemes for the heat equation in dimension one. These schemes are inspired by the Le Potier's trick [CRAS Paris, I 348, 2010]. They preserve the maximum principle and admit a finite volume formulation. We provide a functional setting for the analysis of convergence of such methods. Finally we construct, analyze and test a new explicit non linear maximum preserving scheme: we prove third order convergence: it is optimal on numerical tests

Trace class properties of the non homogeneous linear Vlasov-Poisson equation in dimension 1+1

We consider the abstract scattering structure of the non homogeneous linearized Vlasov-Poisson equations from the viewpoint of trace class properties which are emblematic of the abstract scattering theory [13, 14, 15, 19]. In dimension 1+1, we derive an original reformulation which is trace class. It yields the existence of the Moller wave operators. The non homogeneous background electric field is periodic with 4 + ε bounded derivatives. Mathematics Subject Classification (2010). Primary: 47A40; Secondary: 35P25

Hybrid resonance of Maxwell's equations in slab geometry

Hybrid resonance is a physical mechanism for the heating of a magnetic plasma. In our context hybrid resonance is a solution of the time harmonic Maxwell's equations with smooth coefficients, where the dielectric tensor is a non diagonal hermitian matrix. The main part of this work is dedicated to the construction and analysis of a mathematical solution of the hybrid resonance with the limit absorption principle. We prove that the limit solution is singular: it is constituted of a Dirac mass at the origin plus a principle value and a smooth square integrable function. The formula obtained for the plasma heating is directly related to the singularity.Comment: This published version has been edited to improve the presentation of the result

Angular Momentum preserving cell-centered Lagrangian and Eulerian schemes on arbitrary grids

We address the conservation of angular momentum for cell-centered discretization of compressible fluid dynamics on general grids. We concentrate on the Lagrangian step which is also sufficient for Eulerian discretization using Lagrange+Remap. Starting from the conservative equation of the angular momentum, we show that a standard Riemann solver (a nodal one in our case) can easily be extended to update the new variable. This new variable allows to reconstruct all solid displacements in a cell, and is analogous to a partial Discontinuous Galerkin (DG) discretization. We detail the coupling with a second- order Muscl extension. All numerical tests show the important enhancement of accuracy for rotation problems, and the reduction of mesh imprint for implosion problems. The generalization to axi-symmetric case is detailed

Coupling strategies for compressible - low Mach number flows

International audienceIn order to enrich the modelling of fluid flows, we investigate in this paper a coupling between two models dedicated to distinct regimes. More precisely, we focus on the influence of the Mach number as the low Mach case is known to induce theoretical and numerical issues in a compressible framework. A moving interface is introduced to separate a compressible model (Euler with source term) and its low Mach counterpart through relevant transmission conditions. A global steady state for the coupled problem is exhibited. Numerical simulations are then performed to highlight the influence of the coupling by means of a robust numerical strategy

Dissipative formulation of initial boundary value problems for Friedrichs' systems

International audienceIn this article we present a dissipative definition of a solution for initial boundary value problems for Friedrichs' systems posed in the space L^2_{t,x}. We study the information contained in this definition and prove an existence and uniqueness theorem in the non-characteristic case and with constant coefficients. Finally, we compare our choice of boundary condition to previous works, especially on the wave equation

A one-mesh method for the cell-centered discretization of slide lines

A new method is described to handle slide lines in cell-centered Lagrangian schemes for the modeling of sliding problems between two fluids in the framework of compressible hy- drodynamics. The method is an extension of the one proposed in the reference [1] and is conservative in momentum and total energy. Our method is based on the minimization of an objective function over a specific set that models the sliding constraint. We illustrate on several basic problems