Search

In this note, we introduce a microscopic model for the motion of gas bubbles in a viscous fluid. By interpreting a bubble as a compressible fluid with infinite shear viscosity, we derive a pde/ode system coupling the density/velocity/pressure in the surrounding fluid with the linear/angular velocities and radii of the bubbles. We provide a 1D analogue of the system and construct an existence theory for this simplified system in a natural regularity framework. The second part of the paper is a preparatory work for the derivation of an averaged or macroscopic model.

International audience; 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 relaxation system independent of stiffness of the source term and of the space step. The boundary is approximated using a summation-by-parts method and the stiff stability is proved by energy estimates and Laplace transform. We also investigate if the condition is also necessary, following the continuous case studied by Xin and Xu (2000).; Nous étudions la stabilité du schéma semi-discret centré pour l'équation des ondes linéaire amortie posé sur un demi-espace. Nous dégageons une condition suffisante portant sur la condition de bord, pour la stabilité du problème semi-discret avec donnée initiale et donnée de bord, ceci de manière uniforme par rapport à la raideur du terme source de relaxation ainsi qu'au pas d'espace. La discrétisation de la condition de bord employée provient de l'approche SBP et l'uniforme stabilité s'obtient par l'utilisation de méthodes d'énergie et de la transformée de Laplace. Nous examinons également au travers d'expériences numériques le caractère nécessaire de la condition retenue, de sorte à confronter notre résultat à l'étude de Xin et Xu (2000) portant sur le cas continu.

International audience; The stability of stationary solutions of first-order systems of PDE's are considered. They may include some singular geometric terms, leading to discontinuous flux and non-conservative products. Based on several examples in Fluid Mechanics, we assume that these systems are endowed with a partially convex entropy. We first construct an associated relative entropy which allows to compare two states which share the same geometric data. This way, we are able to prove the stability of some stationary states within entropy weak solutions. This result applies for instance to the shallow-water equations with bathymetry. Besides, this relative entropy can be used to study finite volume schemes which are entropy-stable and well-balanced, and due to the numerical dissipation inherent to these methods, asymptotic stability of discrete stationary solutions is obtained. This analysis does not make us of any specific definition of the non-conservative products, applies to non-strictly hyperbolic systems, and is fully multidimensional with unstructured meshes for the numerical methods.

In this paper, we design and analyze a numerical scheme that approximates a Jin–Xin linear system with implicit equilibrium on a bounded domain. This scheme relaxes toward the asymptotic limit of the linear system. The main properties of the limiting scheme are that it does not require to invert the implicit function defining the manifold and that it provides an accurate discretization of the boundary conditions.

We study a model for compressible multiphase flows involving N non miscible barotopic phases where N is arbitrary. This model boils down to the barotropic Baer-Nunziato model when N = 2. We prove the weak hyperbolicity property, the non-strict convexity of the natural mathematical entropy, and the existence of a symmetric form.

In this article we present a dissipative definition of a solution for initial boundary value problems for Friedrichs’ systems posed in the space . We study the information contained in this definition and prove an existence and uniqueness theorem in the non-characteristic case and with constant coefficients. Finally, we compare our choice of boundary condition to previous works, especially on the wave equation and show how to model additional constrained problems in view of initial boundary value problems for viscoplastic equations.

In this paper, we study a one-dimensional fluid modelled by the Burgers equation influenced by an arbitrary but finite number of particles N(t) moving inside the fluid, each one acting as a point-wise drag force with a particle-related friction constant \(\lambda \). For given particle paths \(h_i(t)\), we only assume finite speed of particles, allowing for crossing, merging and splitting of particles. This model is an extension of existing models for fluid interactions with a single particle; compare (Andreianov et al., SIAM J Math Anal 46(2):1030–1052, 2014, [3], Lagoutiere et al., J Differ Equ 245(11):3503–3544, 2008, [10]): $$ \partial _t u(x,t) + \partial _x \left( \frac{u^2}{2}\right) = \sum _{i=1}^N \lambda (h_i'(t)-u(t,h_i(t))\delta (x-h_i(t))$$ Well-posedness for the Cauchy problem, as well as an \(L^\infty \) bound, is proven under the weak assumption that particle paths are Lipschitz continuous. In this context, an entropy admissibility criteria are shown, using the theory of \(L^1\)-dissipative germs, compare (Andreianov et al., Arch Ration Mech Anal 201:26–86, 2011, [2]), to deal with the moving interfaces resulting from the point-wise particles and the shock waves from the fluid equation interacting with them.

The modeling and the numerical simulation of two-phase flows are investigated for several decades. When dealing with very heterogeneous problems, for instance a water flow with many bubbles, one has to make use of averaged models since the description of each phase and interface is out of reach. Whatever the average is, the resulting models often suffer from severe mathematical pathologies: lack of hyperbolicity, non-conservative products, non-preservation of admissible states... In 1986, Baer and Nunziato proposed an original model which possesses interesting features from the mathematical point of view. Our goal is to provide a (partial) state of art on this model and its derivatives, but also to list some open questions.

International audience; We propose a dynamic model adaptation method for a nonlinear conservation law coupled with an ordinary differential equation. This model, called the ''fine model", involves a small time scale and setting this time scale to 0 leads to a classical conservation law, called the ''coarse model", with a flux which depends on the unknown and on space and time. The dynamic model adaptation consists in detecting the regions where the fine model can be replaced by the coarse one in an automatic way, without deteriorating the accuracy of the result. To do so, we provide an error estimate between the solution of the fine model and the solution of the adaptive method, enabling a sharp control of the different parameters. This estimate rests upon stability results for conservation laws with respect to the flux function. Numerical results are presented at the end and show that our estimate is optimal.

International audience; This paper is devoted to present an approximation of a Cauchy problem for Friedrichs' systems under convex constraints. It is proved the strong convergence in L^2_{loc} of a parabolic-relaxed approximation towards the unique constrained solution.

International audience; We study the finite volume approximation of strong solutions to nonlinear systems of conservation laws. We focus on time-explicit schemes on unstructured meshes, with entropy satisfying numerical fluxes. The numerical entropy dissipation is quantified at each interface of the mesh, which enables to prove a weak–BV estimate for the numerical approximation under a strengthen CFL condition. Then we derive error estimates in the multidimensional case, using the relative entropy between the strong solution and its finite volume approximation. The error terms are carefully studied, leading to a classical $h^1/4$ estimate in $L^2$ under this strengthen CFL condition.

In this paper, we present an original derivation process of a non-hydrostatic shallow water-type model which aims at approximating the incompressible Euler and Navier-Stokes systems with free surface. The closure relations are obtained by a minimal energy constraint instead of an asymptotic expansion. The model slightly differs from the well-known Green-Naghdi model and is confronted with stationary and analytical solutions of the Euler system corresponding to rotational flows. At the end of the paper, we give time-dependent analytical solutions for the Euler system that are also analytical solutions for the proposed model but that are not solutions of the Green-Naghdi model. We also give and compare analytical solutions of the two non-hydrostatic shallow water models.

International audience; In numerous industrial CFD applications, it is usual to use two (or more)different codes to solve a physical phenomenon: where the flow is a priori assumed to have a simple behavior, a code based on a coarse model is applied, while a code based on a fine model is used elsewhere. This leads to a complex coupling problem with fixed interfaces. The aim of the present work is to provide a numerical indicator to optimize to position of these coupling interfaces. In other words, thanks to this numerical indicator, one could verify if the use of the coarser model and of the resulting coupling does not introduce spurious effects. In order to validate this indicator, we use it in a dynamical multiscale method with moving coupling interfaces. The principle of this method is to use as much as possible a coarse model instead of the fine model in the computational domain, in order to obtain an accuracy which is comparable with the one provided by the fine model. We focus here on general hyperbolic systems with stiff relaxation source terms together with the corresponding hyperbolic equilibrium systems. Using a numerical Chapman-Enskog expansion and the distance to the equilibrium manifold, we construct the numerical indicator. Based on several works on the coupling of different hyperbolic models, an original numerical method of dynamic model adaptation is proposed. We prove that this multiscale method preserves invariant domains and that the entropy of the numerical solution decreases with respect to time. The reliability of the adaptation procedure is assessed on various 1D and 2D test cases coming from two-phase flow modeling.

We study the strong stability of one- step explicit finite difference schemes set on the half-space with a left boundary condition. We work on schemes which are consistent with the scalar advection equation. Thanks to Kreiss theorem and GKS theory, the strong stability is equivalent to the absence of zero of the Kreiss–Lopatinskii determinant outside the open unit disk. Then we describe a numerical strategy to count the number of zeros of the Kreiss–Lopatinskii determinant in this domain.The first part deals with different ap- proaches to work on stability and introduce the tools needed to understand the contribu- tions. The second part presents the details of the theoritical results and the numerical strate- gies for the particular case of totally upwind scheme and for the general case.The goal is to introduce and to study a ro- bust and efficient numerical strategy to handle strong stability, thanks to numerical tools and the uniform Kreiss–Lopatinskii condition.; Dans ce manuscrit, nous étudions la stabilité forte des schémas numériques ex- plicites à un pas à coefficients constants, posés sur le demi-espace et possédant un bord à gauche. On suppose que ces sché- mas sont consistants avec l’équation de trans- port scalaire uni-dimensionnelle comportant une donnée de bord à gauche. Grâce au théo- rème de Kreiss et à la théorie développée par Gustafsson, Kreiss et Sundström, la stabi- lité forte est équivalente à l’absence de zéros du déterminant de Kreiss–Lopatinskii à l’exté- rieur du disque unité ouvert. On va alors dé- crire une stratégie numérique permettant de compter les zéros du déterminant de Kreiss– Lopatinskii afin de pouvoir conclure sur la sta- bilité forte du schéma.La première partie de ce manuscrit dé- crit plusieurs approches de la stabilité et in- troduit les objets nécessaires à la compréhen- sion des contributions, notamment la théorie de Gustafsson, Kreiss et Sundström et le dé- terminant de Kreiss–Lopatinskii. La deuxième partie est dédiée aux résultats théoriques et aux stratégies numériques pour le cas particu- lier des schémas totalement décentrés et pour le cas général.L’enjeu est de trouver des stratégies ef- ficaces et robustes pour étudier la stabilité de ces schémas, notamment au travers d’ou- tils numériques et de la condition de Kreiss– Lopatinskii uniforme représentée par le déter- minant de Kreiss–Lopatinskii.

International audience

National audience

This thesis is about theoretical study and numerical treatment of some problems raised in incompressible free-surface fluid dynamics. The first part concerns a model called the Green-Naghdi (GN) equations. Similarly to the non-linear shallow water system (also called Saint-Venant system), the Green--Naghdi equations is a shallow water approximation of water waves problem. Indeed, GN equation is one order higher in approximation compared to Saint-Venant system. For this reason, it contains all the terms this latter system in addition to some non-linear third order dispersive terms. In other words, the GN equation is a dispersive perturbation of the Saint-Venant system. The latter system is hyperbolic and fits the general framework developed in the literature for hyperbolic systems. Particularly, it admits an entropy in the sense of Lax and is symmertizable. Therefore, we can apply the well-posedness results developed for symmetric hyperbolic systems. During the first part of this work, we generalize the notion of symmetry to a more general type of equations including the GN system. This lets us symmetrize the GN equations. Then, we use the suggested symmetric structure to obtain a global existence result for the system with a second order dissipative term, by adapting the approach classically used for hyperbolic systems.The second part of this thesis concerns the numerical treatment of the free surface incompressible Navier-Stokes equation with surface tension. We use the level set formulation to represent the fluid free-surface.Thanks to this formulation, the kinematic boundary condition is treated by solving an advection equation satisfied by the level set function. This equation is solved on a computational domain containing the fluid domain, over small time subintervals. Each iteration of the algorithm corresponds to the adevction of the fluid domain on a small time subinterval and to solve the time-discretized Navier-Stokes equations only on the fluid domain.The time discretization of the Navier--Stokes equation is done by the characteristic method. Then, the key tool which lets us solve this equation on the fluid domain, is an anisotropic mesh adaptation. Indeed, the mesh is adapted at each iteration such that we get convenient approximation errors in the vicinity of the fluid domain. The resolution of the discretized Navier-Stokes equation is done using the Uzawa algorithm for a convenient finite element method. The slip boundary conditions are considered by adding a penalization term to the variational formulation associated to the problem., Cette thèse est consacrée à l’étude théorique ainsi qu’au traitement numérique de fluides incompressibles à surface libre. La première partie concerne un système d’équations appelé le système de Green–Naghdi. Comme le système de Saint-Venant, il s’agit d’une approximation d’eaux peu-profondes du problème de Zakharov. La différence est que le système de Green–Naghdi est d’un degré plus élevé en ordre d’approximation. C’est pourquoi il contient tous les termes du système de Saint-Venant plus de termes d’ordre trois non-linéairement dispersives. Autrement dit, le système de Green–Naghdi peut être vu comme une perturbation dispersive du système de Saint-Venant. Ce dernier système étant hyperbolique, il entre dans le cadre classique développé pour des systèmes hyperboliques. En particulier, il est entropique (au sense de Lax) et symétrique. On peut donc lui appliquer les résultats d’existence et d’unicité bien connus pour des systèmes hyperboliques. Dans la première partie de ce travail, on généralise la notion de symétrie à une classe plus générale de systèmes contenant le système de Green–Naghdi. Ceci nous permet de symétriser les équations de Green–Naghdi et d’utiliser la symétrie obtenue pour déduire un résultat d’existence globale après avoir ajouté un terme dissipative d’ordre 2 au système. Ceci est fait en adaptant l’approche utilisée dans la littérature pour des systèmes hyperboliques.La deuxième partie de ce travail concerne le traitement numérique des équations de Navier–Stokes à surface libre avec un terme de tension de surface. Ici, la surface libre est modélisée en utilisant la formulation des lignes de niveaux. C’est pourquoi la condition cinématique (condition de l’évolution de surface libre) s’écrit sous la forme d’une équation d’advection satisfaite par la fonction de ligne de niveaux. Cette équation est résolue sur une domaine de calcul contenant strictement le domaine de fluide, sur de petits sous-intervalles du temps. Chaque itération de l’algorithme global correspond donc à l’advection du domaine du fluide sur le sous-intervalle du temps associé et ensuite de résoudre le système de Navier–Stokes discrétisé en temps sur le domaine du fluide. Cette discrétisation en temps est faite par la méthode des caractéristiques. L’outil clé qui nous permet de résoudre ce système uniquement sur le domaine du fluide est l’adaptation de maillage anisotrope. Plus précisément, à chaque itération le maillage est adapté au domaine du fluide tel que l’erreur d’approximation et l’erreur géométrique soient raisonnablement petites au voisinage du domaine du fluide. La résolution du problème discrétisé en temps sur le domaine du fluide est faite par l’algorithme d’Uzawa utilisé dans la cadre de la méthode des éléments finis. Par ailleurs, la condition de glissement de Navier est traité ici en ajoutant un terme de pénalisation à la formulation variationnelle associée.

In this thesis, we consider constrained hyperbolic partial differential equations and more precisely mechanical problems coming from perfect plasticity. The goal of this thesis is to study these problems thanks to different approaches, to analyze the interactions between these different points of view and to confront these various analyzes to get new results. A brief review of the mechanical origin of perfect plasticity problems and also of the previous results on these topics are described in Chapter 1.In Chapter 2, we focus our attention on hyperbolic systems with boundary conditions. First, we develop a weak theory for these problems and explain, in a simplified case, why this theory is well-posed. Then, we introduce similarly a notion of weak solutions for constrained hyperbolic systems with boundary conditions.Chapter 3 is devoted to the study of the simplified model of dynamical perfect plasticity. We confront the approach introduced in the previous chapter with the one, more standard, coming from calculus of variations that allows us to obtain existence and uniqueness of the solutions for this model. It allows us to bring to light a new interaction between the boundary conditions and the constraints and to get a short-time regularity theorem.Lastly, in Chapter 4, we are interested in the numerical approximation of constrained hyperbolic systems thanks to finite volume schemes. This work allows us to get a convergence result for problems without boundary condition and to show numerically the link between boundary conditions and constraints on the example of the previous chapter., Dans cette thèse, nous nous intéressons aux équations aux dérivées partielles hyperboliques sous contraintes ; plus particulièrement aux problèmes provenant de la mécanique de la plasticité parfaite. L'objectif de cette thèse est d'étudier ces problèmes via différentes approches, d'analyser les interactions entre ces points de vues distincts et de tirer profit de ces analyses différentes pour obtenir de nouveaux résultats. Un bref historique de l'origine mécanique des problèmes de la plasticité parfaite ainsi que des résultats précédemment obtenus sont décrits dans le Chapitre 1.Dans le Chapitre 2, nous concentrons notre attention sur les systèmes hyperboliques avec conditions de bord. Dans un premier temps, nous développons une théorie faible pour ces problèmes et expliquons dans un cas simplifié le caractère bien posé de cette théorie. Puis, nous introduisons de manière similaire la notion de solution faible pour des systèmes hyperboliques avec condition de bord soumis à une contrainte.Le Chapitre 3 est dédié à l'étude d'un modèle simplifié de la dynamique de la plasticité parfaite. Nous confrontons l'approche introduite au chapitre précédent avec celle, plus classique, provenant du calcul des variations qui permet d'obtenir l'existence et l'unicité des solutions pour ce modèle. Cela nous permet de mettre en évidence une nouvelle interaction entre les conditions de bord et les contraintes ainsi que d'aboutir à un théorème de régularité des solutions en temps courts.Enfin, dans le Chapitre 4, nous nous intéressons à l'approximation numérique des systèmes hyperboliques sous contraintes grâce à des schémas de type volumes finis. Ce travail nous permet d'obtenir un résultat de convergence pour les problèmes sans bord et d'illustrer numériquement les interactions entre les conditions de bord et les contraintes sur l'exemple du chapitre précédent.