Degenerate Anisotropic Elliptic Problems and Magnetized Plasma Simulations

This paper is devoted to the numerical approximation of a degenerate anisotropic elliptic problem.The numerical method is designed for arbitrary spacedependent anisotropy directions and does not require any specially adapted coordinate system.It is also designed to be equally accurate in the strongly and the mildly anisotropic cases.The method is applied to the Euler-Lorentz system,in the drift-fluid limit.This system provides a model for magnetized plasmas.

Thin Layer Models for Electromagnetism

We present a review on the accuracy of asymptotic models for the scattering problem of electromagnetic waves in domains with thin layer.These models appear as first order approximations of the electromagnetic field.They are obtained thanks to a multiscale expansion of the exact solution with respect to the thickness of the thin layer,that makes possible to replace the thin layer by approximate conditions.We present the advantages and the drawbacks of several approximations together with numerical validations and simulations.The main motivation of this work concerns the computation of electromagnetic field in biological cells.The main difficulty to compute the local electric field lies in the thinness of the membrane and in the high contrast between the electrical conductivities of the cytoplasm and of the membrane,which provides a specific behavior of the electromagnetic field at low frequencies.

Relaxation Schemes for the M1 Model with Space-Dependent Flux:Application to Radiotherapy Dose Calculation

Because of stability constraints,most numerical schemes applied to hyperbolic systems of equations turn out to be costly when the flux term is multiplied by some very large scalar.This problem emerges with the M_(1)system of equations in the field of radiotherapy when considering heterogeneous media with very disparate densities.Additionally,the flux term of the M_(1)system is non-linear,and in order for the model to be well-posed the numerical solution needs to fulfill conditions called realizability.In this paper,we propose a numerical method that overcomes the stability constraint and preserves the realizability property.For this purpose,we relax the M_(1)system to obtain a linear flux term.Then we extend the stencil of the difference quotient to obtain stability.The scheme is applied to a radiotherapy dose calculation example.

Mathematical analysis of an HIV infection model including quiescent cells and periodic antiviral therapy

In this paper, we revisit the model by Guedj et al. [J. Guedj, R. Thibaut and D. Commenges, Maximum likelihood estimation in dynamical models of HIV, Biometrics 63 （2007） 198-206; J. Guedj, R. Thibaut and D. Commenges, Practical identifiability of HIV dynamics models, Bull. Math. Biol. 69 （2007） 2493 2513] which describes the effect of treatment with reverse transcriptase （RT） inhibitors and incorporates the class of quiescent cells. We prove that there is a threshold value 7 of drug efficiency η such that if η 〉 7, the basic reproduction number R0 〈 1 and the infection is cleared and if η〈 η^-, the infectious equilibrium is globally asymptotically stable. When the drug efficiency function η（t） is periodic and of the bang-bang type we establish a threshold, in terms of spectral radius of some matrix, between the clearance and the persistence of the disease. As stated in related works [L. Rong, Z. Feng and A. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment, Bull. Math. Biol. 69 （2007） 2027-2060; P. De Leenheer, Within-host virus models with periodic antiviral therapy, Bull. Math. Biol. 71 （2009） 189-210.], we prove that the increase of the drug efficiency or the active duration of drug must clear the infection more quickly. We illustrate our results by some numerical simulations.

Global hydrostatic approximation of the hyperbolic Navier-Stokes system with small Gevrey class 2 data In memory of Professor Geneviève Raugel

We investigate the hydrostatic approximation of a hyperbolic version of Navier-Stokes equations,which is obtained by using the Cattaneo type law instead of the Fourier law,evolving in a thin strip R×(0,ε).The formal limit of these equations is a hyperbolic Prandtl type equation.We first prove the global existence of solutions to these equations under a uniform smallness assumption on the data in the Gevrey class 2.Then we justify the limit globally-in-time from the anisotropic hyperbolic Navier-Stokes system to the hyperbolic Prandtl system with such Gevrey class 2 data.Compared with Paicu et al.(2020)for the hydrostatic approximation of the 2-D classical Navier-Stokes system with analytic data,here the initial data belongs to the Gevrey class 2,which is very sophisticated even for the well-posedness of the classical Prandtl system(see Dietert and GerardVaret(2019)and Wang et al.(2021));furthermore,the estimate of the pressure term in the hyperbolic Prandtl system gives rise to additional difficulties.

Numerical Simulations of Rarefied Gases in Curved Channels: Thermal Creep, Circulating Flow, and Pumping Effect

We present numerical simulations of a new system of micro-pump based on the thermal creep effect described by the kinetic theory of gases.This device is made of a simple smooth and curved channel with a periodic temperature distribution.Using the Boltzmann-BGK model as the governing equation for the gas flow,we develop a numerical method based on a deterministic finite volume scheme,implicit in time,with an implicit treatment of the boundary conditions.This method is comparatively less sensitive to the slow flow velocity than the usual Direct Simulation Monte Carlo method in case of long devices,and turns out to be accurate enough to compute macroscopic quantities like the pressure field in the channel.Our simulations show the ability of the device to produce a one-way flow that has a pumping effect.

AN EXPONENTIAL INEQUALITY FOR AUTOREGRESSIVE PROCESSES IN ADAPTIVE TRACKING

A wide range of literature concerning classical asymptotic properties for linear models with adaptive control is available, such as strong laws of large numbers or central limit theorems. Unfortunately, in contrast with the situation without control, it appears to be impossible to find sharp asymptotic or nonasymptotic properties such as large deviation principles or exponential inequalities. Our purpose is to provide a first step towards that direction by proving a very simple exponential inequality for the standard least squares estimator of the unknown parameter of Gaussian autoregressive process in adaptive tracking.

A Residual Distribution Method Using Discontinuous Elements for the Computation of Possibly Non Smooth Flows

In this paper,we describe a residual distribution(RD)method where,contrarily to“standard”this type schemes,the mesh is not necessarily conformal.It also allows to use discontinuous elements,contrarily to the“standard”case where continuous elements are requested.Moreover,if continuity is forced,the scheme is similar to the standard RD case.Hence,the situation becomes comparable with the Discontinuous Galerkin(DG)method,but it is simpler to implement than DG and has guaranteed L^(∞)bounds.We focus on the second-order case,but the method can be easily generalized to higher degree polynomials.

A Review of Residual Distribution Schemes for Hyperbolic and Parabolic Problems:The July 2010 State of the Art

We describe and review non oscillatory residual distribution schemes that are rather natural extension of high order finite volume schemes when a special emphasis is put on the structure of the computational stencil.We provide their connections with standard stabilized finite element and discontinuous Galerkin schemes,show that their are really non oscillatory.We also discuss the extension to these methods to parabolic problems.We also draw some research perspectives.