## 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.

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.

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, 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.

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.

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 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.

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