Les publications sont usuellement référencées, de façon plus ou moins complète, sur les sites suivants.
Référencement bibliographique: MathScinet Zbmath
Finite difference methods for hyperbolic problems with boundaries: stability and multiscale analysis
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 …
On the stability of totally upwind schemes for the hyperbolic initial boundary value problem
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 …
High order asymptotic preserving scheme for linear kinetic equations with diffusive scaling
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 …
Stability of finite difference schemes for the hyperbolic initial boundary value problem by winding number computation
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 …
Modified lawson methods for Vlasov equations
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 …
High order numerical schemes for transport equations on bounded domains
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 …
Coupling techniques for nonlinear hyperbolic equations. II. Resonant interfaces with internal structure
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 …
A stiffly stable fully discrete scheme for the damped wave equation using discrete transparent boundary condition
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 …
A stiffly stable semi-discrete scheme for the characteristic linear hyperbolic relaxation with boundary
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 basal level of gene expression associated with chromatin loosening shapes Waddington landscapes and controls cell differentiation
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 …
Stability of finite difference schemes for hyperbolic initial boundary value problems: numerical boundary layers
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 …
Some remarks about flows of Hilbert-Schmidt operators
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 accuracy of biochemical interactions is ensured by endothermic stepwise kinetics
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 …
Shock profiles for the Shallow-water Exner model
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, …
Coupling techniques for nonlinear hyperbolic equations. IV. Well-balanced schemes for scalar multi-dimensional and multi-component laws
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 …
Coupling techniques for nonlinear hyperbolic equations
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 …
Coupling techniques for nonlinear hyperbolic equations. III. The well-balanced approximation of thick interfaces.
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 …
Extension of ALE methodology to unstructured conical meshes
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 …
Coupling techniques for nonlinear hyperbolic equations. I. Self-similar diffusion for thin interfaces
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 …
A second order anti-diffusive Lagrange-remap scheme for two-component flows
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 …
Existence result for the coupling problem of two scalar conservation laws with Riemann initial data
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 $x = 0$ …
Étude mathématique et numérique d'équations hyperboliques non-linéaires : couplage de modèles et chocs non classiques
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 …
Fluid-particles flows: a thin spray model with energy exchanges
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 …
Dafermos regularization for interface coupling of conservation laws
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 …
Coupling Two Scalar Conservation Laws via Dafermos' Self-Similar Regularization
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. …
Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I
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 …
A Sharp Interface and Fully Conservative Scheme for Computing Nonclassical Shocks
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 …