## Institut de Mathématiques |
## European Union - H2020 Program |
## University Paul Sabatier |

We are interested in the analysis of a numerical scheme discretizing drift-diffusion systems for semiconductors. The considered scheme is finite volume in space, and the numerical fluxes are a generalization of the classical Scharfetter-Gummel scheme, which allows to consider both linear or nonlinear pressure laws. Using a discrete entropy method, we establish the convergence of approximate solutions towards an approximation of the thermal equilibrium state as time tends to infinity. We obtain an exponential decay rate by controlling the discrete relative entropy with the entropy production. This result is proved under assumptions of existence and uniform-in-time upper and lower bounds for numerical solutions, which will be obtained in the linear case thanks to an adaptation to the discrete framework of the Moser's iteration technique.

: For a linear system of PDEs, Trefftz Discontinuous Galerkin (TDG) methods yield a systematic way to introduce exact solutions in the approximation space. This strategy is appeling for transport equations with stiff absorption/scattering coefficients, since it allows to introduce some "Caseology" in the approximation space. We use this method for the discretization of PN systems. Among the main multiD results, we will concentrate on the well-posedness, the AP property and the ability to capture boudary layers. Numerical results confirm the theoretical results.

It is important for the applications to reentry in the atmosphere of a vehicle to be able to establish a precise link between the rarefied regime where the Boltzmann equation is used and the fluid regime where the Navier-Stokes equations can be considered. However, due to the presence of diatomic molecules O_2, N_2, and to monoatomic species like O, N or Ar around the shuttle, one must deal with a mixture of monoatomic and polyatomic gases. We explain the difficulties related to this issue when one tries to perform the Chapman Enskog asymptotics, and how it can be solved when simple models of internal energy transfer are taken into account. For such models, a complete computation of the viscosity coefficients can be performed.

This talk explores the regularization properties of the linear Fokker-Planck equations, with a diffusion (in velocity) matrix field that is measurable, bounded and uniformly elliptic. By adapting the DeGiorgi method for elliptic equations with rough coefficients, we prove that weak solutions of the Fokker-Planck equation satisfying the energy inequality and the renormalization property are in fact Hölder continuous. This is an account of a joint work in collaboration with C. Imbert (Ecole normale sup\'erieure, Paris), C. Mouhot (University of Cambridge) and A. Vasseur (University of Texas, Austin).

I will present results obtained in collaboration with Nicolas Fournier, on the propagation of chaos for the Landau equation. The difficulty here is the presence of a singularity in the interaction kernel that appears in equation. For mild singularities, we obtain quantitative results of convergence using a weak-strong stability result for the Landau equation, and a perturbation of it, that allows to apply it also to empirical measures associated to particles system approximating the Landau equation. For stronger singularity, we obtain a qualitative result of convergence, relying on the techniques introduced previously with Stéphane Mischler for the case of vortex, but with several improvement in order to control the possibly degenerate Landau diffusion.

In this talk, I will present results obtained in collaboration with L. Miguel Rodrigues. We consider a plasma of electrons in an inhomogeneous background of ions. We are interested in the dynamics of the light particles which is modeled by the Vlasov-Poisson-Fokker-Planck equation. In the appropriate scaling where characteristic time scales are those of the ions, an important dimensionless parameter appears, the mass ratio between an electron and an ion. Our focus is on deriving an asymptotic model when the mass ratio tends to $0$. In this regime, strong anisotropic phenomena occurs; while adiabatic equilibrium along magnetic field lines is asymptotically reached our limit model captures a non trivial guiding-center dynamics in the perpendicular directions. We do check that in any case the obtained asymptotic model defines a well-posed dynamical system and when self consistent electric fields are neglected we provide a rigorous mathematical justification of the formally derived systems. In this last step we provide a complete control on solutions by developing anisotropic hypocoercive estimates.

The talk will present a numerical method for a hierarchy of equations ranging from (stochastic) interacting particle systems, over nonlinear mean-field or Fokker-Planck equations, to hydrodynamic macroscopic equations with non-local terms. We discuss an ALE-method based on a mesh-free particle discretization. An asymptotic preserving scheme for the transition between the equations of the hierarchy is discussed. The scheme can be viewed as a numerical transition from a microscopic Discrete-Element simulation (DEM) to a mesh-free method for fluid dynamic equations. Several numerical test cases for diffusive and hyperbolic limits are presented. Finally, applications to pedestrian flow simulations are shown.

We will discuss the choice of the discretizations of the differential operators appearing in the Fokker-Planck equation to ensure good numerical properties of the long time behavior, in the spirit of the already known results of hypocoercivity and exponential time decay in the continuous setting.

Our aim is to construct efficient numerical approximations to solve a class of highly oscillatory evolution problems. In the case of time oscillations, we will briefly present some averaging techniques and show how to derive high order asymptotic models. Then we present two new strategies to construct numerical schemes having the important property of being uniformly accurate (UA) with respect to the frequency. Both the two approaches boil down the highly oscillatory problem to uniformly smooth dynamics with respect to the frequency, which allows the construction of UA numerical scheme straightforwardly. Extensions to time-space oscillations are shown to be possible in some cases, and applications in quantum mechanics and kinetic theory will be presented.

In this talk, I will present some recent works done in collaboration with François Delarue, Benoît Fabrèges, Hélène Hivert, Kevin Le Balc'h, Sofiane Martel and Nicolas Vauchelet, on discrete algorithms for the aggregation equations with pointy potentials (admitting measure solutions). The results are of two different types. In a first part, I will prove that first order upwind typed schemes are convergent to the 1/2 order (in the space and time steps) in Wasserstein distance, in any space dimension on Cartesian grids. In a second part, I will present some extensions of these schemes, either on non-Cartesian grids or with better accuracy.

We will report our effort in developing and analyzing high order asymptotic preserving methods for some linear kinetic models in a diffusive scaling. The methods are based on reformulated forms of the equations, and involve (local) discontinuous Galerkin (DG) spatial discretizations with suitably chosen numerical fluxes, as well as globally stiffly accurate implicit-explicit (IMEX) Runge-Kutta temporal discretizations. One key ingredient in theoretical analysis is the uniform stability, which will be discussed, along with the error estimates, rigorous asymptotic analysis, and some numerical results.

In this talk, we investigate the large time behavior of a agent based model modeling tumor growth. This microscopic model combines short-range repulsion and cell division. We derive the associated macroscopic dynamics leading to a porous media type equation. In order to capture the long-time behavior of the microscopic model, we have to modify the porous media in order to include a density threshold for the repulsion. The main difficulty is then to investigate the limit as the repulsion between cells becomes singular (modeling non-overlapping constraint). We show formally that such asymptotic limit leads to a free-boundary problem (Hele-Shaw type). Numerical results confirm the relevance of such limit.

Many problems in nature exhibit multiscale behaviours, which are very difficult to treat from an analytical point of view, but also from a numerical point of view. Standard explicit numerical techniques require refined meshes (due to stability issues), and thus excessive simulation times. Fully implicit schemes are sometimes also not an issue. I will present in this talk some simple examples of singularly-perturbed problems and their Asymptotic-Preserving resolution, in order to underline the specific features of AP-schemes.

In this talk we will present some recent results on the construction of efficient Monte Carlo methods for uncertainty quantification in kinetic equations. The method is based on a suitable micro-macro decomposition strategy combined with the use of different sampling scales on the random field in order to reduce the variance of standard Monte Carlo sampling methods. Applications to socio-economy and rarefied gas dynamics are presented.

Viscous momentum, heat, mass, and charge transport are traditionally modeled by second-order parabolic partial differential equations which rely on the phenomenological laws of Newton, Fourier, and Fick, respectively (the resistivity is modeled in a similar parabolic way). However, it is possible to describe these processes in a pure hyperbolic framework as will be demonstrated in this talk. Such a hyperbolic approach for describing transport phenomena gains some attractive features from the numerical viewpoint, but also, it allows to describe non-equilibrium transport such as flows of rarefied gases, non-Fourier heat conduction, and non-Fickian diffusion and thus, it may establish a connection between kinetic theory and fluid dynamics. In this talk, we shall present the hyperbolic relaxation framework and demonstrate its applicability in a series of numerical test cases.

We compute an upper bound for the difference between the solution of a gyro-kinetic equation and the exact distribution of gyro-centers, obtained from the Vlasov equation in a non-homogeneous background field, with time-dependent electromagnetic perturbations. The gyro-kinetic equation is derived by averaging the characteristics of Vlasov on the level of the Lagrangian via a coordinate map, the gyro-transformation. This approach due to Littlejohn is called structured averaging, since it preserves the Hamiltonian structure of the characteristics. We prove the existence of such a coordinate map and compute it explicitly for different strengths of variations of the background field.

In this paper, we develop a high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for nonlinear Vlasov-Poisson (VP) simulations without operator splitting. In particular, we combine two recently developed novel techniques: one is the high order non-splitting SLDG transport method \emph{[Cai, et al., J Sci Comput, 2017]}, and the other is the high order characteristics tracing technique proposed in \emph{[Qiu and Russo, J Sci Comput, 2017]}. The proposed method with up to third order accuracy in both space and time is locally mass conservative, free of splitting error, positivity-preserving, stable and robust for large time stepping size. The SLDG VP solver is applied to classic benchmark test problems such as Landau damping and two-stream instabilities for VP simulations. Efficiency and effectiveness of the proposed scheme is extensively tested. Tremendous CPU savings are shown by comparisons between the proposed SL DG scheme and the classical Runge-Kutta DG method.

In many cases, even in highly oscillating dynamics do persist quantities that evolve on slow scales and uncouple at leading order from faster scales. Asymptotically their evolution obeys a closed system of averaged equations. This notably occurs in the dynamics of plasmas subject to a strong external magnetic field. In the latter case the task of untangling a slow dynamics is part of the gyrokinetic theory, taken in a broad sense. I will discuss some first contributions, obtained with Francis Filbet (Toulouse, IUF), that aim at designing numerical schemes that, in this context, do capture the slow part of the dynamics even when discretization meshes are too coarse to describe co-existing fast oscillations.

In this joint work with W. Melis and G. Samaey, we develop a high-order, fully explicit, (almost-) asymptotic-preserving projective integration scheme for nonlinear collisional kinetic equations such as the BGK and Boltzman ones. The method first takes a few small (inner) steps with a simple, explicit method to damp out the stiff components of the solution. Then, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. Based on the spectrum of the linearized collision operator, we deduce that, with an appropriate choice of inner step size, the time step restriction on the outer time step as well as the number of inner time steps is independent of the stiffness of the collisional source term. I will illustrate the method with numerical results in one and two spatial dimensions of space and velocity, using various schemes to compute the transport and collision parts.

It is well known that semi-discrete high order discontinuous Galerkin (DG) methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws and symmetric hyperbolic systems, in any space dimension and for any triangulations. However, this property holds only for the square entropy and the integrations in the DG methods must be exact. It is significantly more difficult to design DG methods to satisfy entropy inequalities for a non-square convex entropy, and / or when the integration is approximated by a numerical quadrature. In this talk, we report on our recent development of a unified framework for designing high order DG methods which will satisfy entropy inequalities for any given single convex entropy, through suitable numerical quadrature which is specific to this given entropy. Our framework applies from one-dimensional scalar cases all the way to multi-dimensional systems of conservation laws. For the one-dimensional case, our numerical quadrature is based on the methodology established in the literature, with the main ingredients being summation-by-parts (SBP) operators derived from Legendre Gauss-Lobatto quadrature, the entropy stable flux within elements, and the entropy stable flux at element interfaces. We then generalize the scheme to two-dimensional triangular meshes by constructing SBP operators on triangles based on a special quadrature rule. A local discontinuous Galerkin (LDG) type treatment is also incorporated to achieve the generalization to convection-diffusion equations. Numerical experiments will be reported to validate the accuracy and shock capturing efficacy of these entropy stable DG methods. This is a joint work with Tianheng Chen.

This work is a collaboration with Giacomo Dimarco (Ferrara), Raphaèle Loubère (Bordeaux) and Victor Michel-Dansac (Toulouse). In this talk, I will present an implicit explicit total variation diminishing methods for the compressible isentropic Euler system in the low Mach number regime. The presented scheme is asymptotically stable with a CFL condition independent from the Mach number and it degenerates, in the low Mach number regime, to a consistent discretization of the incompressible system. Since, it has been proved by S. Gottlieb, C.-W. Shu and E. Tadmor, that implicit schemes of order higher than one cannot be TVD, we construct a new paradigm of implicit time integrators by coupling first order in time schemes with second order ones in the same spirit as highly accurate shock capturing TVD methods in space. The TVD property is proved on a linear model advection equation and then extended to the isentropic Euler case. The result is a method which interpolates from the first to the second order both in space and time, which preserves the monotonicity of the solution, highly accurate for all choices of the Mach number and with a time step only restricted by the non stiff part of the system. Then, we show with one and two dimensional test cases that the method indeed possesses the claimed properties.

In strong magnetized plasma, the 6D Vlasov-Maxwell system can be reduced to a 4D drift-kinetic model, by averaging over the gyroradius of charged particles and assuming a uniform external magnetic field. In the plane perpendicular to the magnetic field, the plasma is governed by the 2D guiding-center model. In this talk, we will start from a 5D Vlasov-Maxwell system (2D in space and 3D in velocity), which incorporates both self-consistent magnetic field and external strong magnetic field, to formally derive a 3D asymptotic model (2D in space and 1D in velocity). Several numerical experiments with high order approximation of the asymptotic model are performed, which provide a solid validation and illustrate the effect of the self-consistent magnetic field on the current density.

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Last update: Jul. 4th 2017