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This habilitation manuscript gathers the work I have done in recent years. They mainly focus on the study of the stability and the multiscale analysis of finite difference methods for the approximation of linear hyperbolic problems with boundaries. In such a context, various scales are likely to be …

In this paper, we present a numerical strategy to check the strong stability (or GKS-stability) of one-step explicit totally upwind schemes in 1D with numerical boundary conditions. The underlying approximated continuous problem is the one-dimensional advection equation. The strong stability is …

In this work, high order asymptotic preserving schemes are constructed and analysed for kinetic equations under a diffusive scaling. The framework enables to consider different cases: the diffusion equation, the advection-diffusion equation and the presence of inflow boundary conditions. Starting …

In this paper, we present a numerical strategy to check the strong stability (or GKS-stability) of one-step explicit finite difference schemes for the one-dimensional advection equation with an inflow boundary condition. The strong stability is studied using the Kreiss-Lopatinskii theory. We …

In this work, Lawson type numerical methods are studied to solve Vlasov type equations on a phase space grid. These time integrators are known to satisfy enhanced stability properties in this context since they do not suffer from the stability condition induced from the linear part. We introduce …

This article is an account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows. The goal is to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible …

In the first part of this series, an augmented PDE system was introduced in order to couple two nonlinear hyperbolic equations together. This formulation allowed the authors, based on Dafermos’s self-similar viscosity method, to establish the existence of self-similar solutions to the coupled …

We study the stability analysis of the time-implicit central differencing scheme for the linear damped wave equation with boundary. Xin and Xu (J. Differential Equations 2000) prove that the initial-boundary value problem (IBVP) for this model is well-posed, uniformly with respect to the stiffness …

We study the stability of the semi-discrete central scheme for the linear damped wave equation with boundary. We exhibit a sufficient condition on the boundary to guarantee the uniform stability of the initial boundary value problem for the relaxation system independently of the stiffness of the …

The baseline level of transcription, which is variable and difficult to quantify, seriously complicates the normalization of comparative transcriptomic data, but its biological importance remains unappreciated. We show that this currently neglected ingredient is essential for controlling gene …

In this article, we give a unified theory for constructing boundary layer expansions for discretized transport equations with homogeneous Dirichlet boundary conditions. We exhibit a natural assumption on the discretization under which the numerical solution can be written approximately as a …

This paper deals with bracket flows of Hilbert–Schmidt operators. We establish elementary convergence results for such flows and discuss some of their consequences.

The discerning behavior of living systems relies on accurate interactions selected from the lot of molecular collisions occurring in the cell. To ensure the reliability of interactions, binding partners are classically envisioned as finely preadapted molecules, selected on the basis of their …

This article is devoted to analyze some ambiguities coming from a class of sediment transport models. The models under consideration are governed by the coupling between the shallow-water and the Exner equations. Since the PDE system turns out to be an hyperbolic system in non conservative form, …

This series of papers is devoted to the formulation and the approximation of coupling problems for nonlinear hyperbolic equations. The coupling across an interface in the physical space is formulated in terms of an augmented system of partial differential equations. In an earlier work, this strategy …

We analyze the coupling between different nonlinear hyperbolic equations across possibly resonant interfaces. The proposed reformulation of the problem involves a nonconservative product that is understood through a self-similar viscous approximation. We obtain the existence of a coupled solution to …

We continue our analysis of the coupling between nonlinear hyperbolic problems across possibly resonant interfaces. In the first two parts of this series, we introduced a new framework for coupling problems which is based on the so-called thin interface model and uses an augmented formulation and an …

We propose a bi-dimensional finite volume extension of a continuous ALE method on unstructured cells whose edges are parameterized by rational quadratic Bezier curves. For each edge, the control point possess a weight that permits to represent any conic (see for example [LIGACH]) and thanks to …

We investigate various analytical and numerical techniques for the coupling of nonlinear hyperbolic systems and, in particular, we introduce an augmented formulation that allows for the modelling of the dynamics of interfaces between fluid flows. The main technical difficulty to be overcome lies in …

We build a non-dissipative second order algorithm for the approximate resolution of the one-dimensional Euler system of compressible gas dynamics with two components. The considered model was proposed in [Allaire-Clerc-Kokh 2002]. The algorithm is based on [Kokh-Lagoutière 2010] which deals with a …

This paper is devoted to the coupling problem of two scalar conservation laws through a fixed interface located for instance at $x = 0$. Each scalar conservation law is associated with its own (smooth) flux function and is posed on a half-space, namely $x < 0$ or $x > 0$. At interface $x = 0$ …

Cette thèse concerne l’étude mathématique et numérique d’équations aux dérivées partielles hyperboliques non-linéaires. Une première partie traite d’une problématique émergente: le couplage d’équations hyperboliques. Les applications poursuivies relèvent du couplage …

This paper is devoted to an asymptotic analysis of a fluid-particles coupled model, in the bubbling regime. On the theoretical point of view, we extend the analysis done in [Carrillo-Goudon 2008] for the case of an isentropic gas to the case of an ideal gas, thus adding the internal energy, or …

We study the coupling of two conservation laws with different fluxes at the interface $x=0$. The coupling condition yields (whenever possible) continuity of the solution at the interface. Thus the coupled model is not conservative in general. This gives rise to interesting questions such as …

We are interested in the problem of coupling two scalar conservation laws with distinct flux-functions. This problem arises, for instance, in modeling fluid flows in media with discontinuous porosity and has important possible applications in the numerical computation of a singular pressure drop. …

We propose a new numerical approach to compute nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces, contrary to standard finite difference schemes. The main challenge …

We present a sharp interface and fully conservative numerical strategy for computing nonclassical solutions of scalar conservation laws. The difficult point is to impose at the discrete level a prescribed kinetic relation along each nonclassical discontinuity. Our method is based on a relevant …