Dynamic processes in continuum physics are modeled using time-dependent partial differential equations (PDE), which are based on the conservation of some physical quantities, such as mass, momentum and energy. Depending on the physical phenomenon under consideration, the governing equations can exhibit some mathematical structures like differential constraints, algebraic relations, physical admissible states as well as asymptotic limits and thermodynamics compatibility. An interesting class of mathematical models is provided by symmetric hyperbolic systems that intrinsically imply all the structures listed above. When passing at the discrete level, the exact satisfaction of these structural properties is not automatically guaranteed, thus Structure Preserving numerical schemes have recently emerged with the aim of exactly discretizing at least a subset of these constraints. We will investigate and present some of our research activity carried out in the framework of the development of Structure Preserving schemes, focusing on recent contributions delivered in the last three years. In particular, we will address asymptotic preserving schemes for low Mach flows, div-curl and curl-grad preserving operators for discontinuous Galerkin methods, and a novel geometric and thermodynamically compatible finite volume method for continuum mechanics.
The energy of saline gradients is a very promising source of non-intermittent renewable energy, the exploitation of which is hampered by the lack of economically viable technology. The most investigated harvesting methods rely on selective transport of ions or water molecules through semi-permeable or ion-selective membranes, which demonstrate limited power densities of the order of a few W/m2. While in the last decade single nanofluidic objects such as nanopores of nanotubes have opened up very promising prospects with power density capabilities in the kW or even MW/m2, scale-up efforts face serious issues, as concentration polarization phenomena result in a massive loss of performance.
At the LiPhy we work on a concept of nanofluidic exchanger for power generation from saline gradients, focused on designing a nanoscale flow able to harvest the power at the output of the nanopores. We will present the study of a simple exchanger made of a selective nanoslit fed by a nanofluidic assembly. One specific feature of such an exchanger relies on the non-linear ion fluxes through the nanoslit, according to the so-called 1D Poisson-Nernt Planck equations. Such an elemental brick could be massively parallelized in stackable electricity-generating layers using standard technologies of the semi-conductors industry. We demonstrate here a scheme for rationalizing the choice of the exchanger parameters, taking into account the transport properties at all scales. The simplified numerical resolution of the three-dimensional device shows that net power densities of 300 W/m2 and more can be achieved.
Orateur(e)s :
Walter Boscheri (LAMA); Camille Carvalho, (ICJ); Frédérique Charles, (LJK); Nicolae Cindea, (LMBP); Sue Claret, (LMBP); Baptiste Devyver, (IF); Martin Donati, (IF); Louis Dupaigne, (ICJ); Hugo Eulry, (UMPA); Christophe Lacave, (LAMA); Mickael Nahon, (LJK); Niami Nasr, (ICJ); Pierre-Damien Thizy, (ICJ);
Orateur(e)s :
Walter Boscheri (LAMA); Camille Carvalho, (ICJ); Frédérique Charles, (LJK); Nicolae Cindea, (LMBP); Sue Claret, (LMBP); Baptiste Devyver, (IF); Martin Donati, (IF); Louis Dupaigne, (ICJ); Hugo Eulry, (UMPA); Christophe Lacave, (LAMA); Mickael Nahon, (LJK); Niami Nasr, (ICJ); Pierre-Damien Thizy, (ICJ);
Je présenterai quelques avancées récentes sur la caractérisation des phénomènes de propagation qui apparaissent dans les équations semi-linéaires avec diffusion non locale de type Levy. Récemment différentes dichotomies entre propagation accélérée et propagation à vitesse constante en fonction des paramètres de décroissance du noyau et de l'ordre d'annulation en zéro de la non-linéarité considérée ont été obtenues. Je me concentrerai sur le cas monostable et sur une manière de contourner les difficultés liées au traitement des opérateurs de Levy généraux.
Let ϒ be a generalised cone in Rd. Roughly speaking, Yaglom limit describes the behaviour of the process conditioned not to exit the cone, or, in other words, not to become extinct or not to be absorbed. In the talk we will discuss the existence of this limit for a class of (not necessarily symmetric) α-stable Lévy processes living in the cone ϒ. To this end, we will use the so-called Martin kernel at inifinty - the invariant function for the killed semigroup - to obtain the so-called entrance law from the origin, which we also call the self-similar solution. Using this approach, for the isotropic case we will also obtain the large-time asymptotics for the killed semigroup and provide several examples of our resutts.
When discretizing partial differential equations, one can choose local (finite differences, volumes, elements) or global (spectral) methods. The most common spectral basis is built on trigonometric polynomials, i.e. Fourier series. It constrains the boundary conditions to be periodic and has been an important tool in physics, used for instance to study theoretical scalings of turbulence. While spectral methods show "exponential convergence" for smooth functions, large DNS simulations also become too expensive for e.g. when reaching very large Reynolds numbers. In practice, it is possible to solve a coarser version of the DNS by removing the largest wavenumbers in spectral space (cut-off) and modeling transfers at the smallest (sub-grid) scales instead. The definition of such a model has been an open problem for a long time and classical ones are either too diffusive or unstable. Machine learning started to be an interesting alternative few years ago and people quickly found that learning a model that performs better on a priori (instantaneous) metrics is possible. We have shown that in order to lead to stable simulations in a posteriori tests, the temporal dimension must be taken into account during the learning process. This problem has now been largely explored with periodic boundary conditions, but when it comes to spectral methods with orthogonal polynomials and fixed boundaries, new challenges appear.
In this work in collaboration with T. Iguchi, we show that the Saint-Venant equations in 2D with a partially submerged obstacle is well-posed. To do so, we show that the problem is equivalent to the usual Saint-Venant equations in an external domain, with additionnal non-standard boundary conditions because they are not local in space and time. These conditions do not fit into any category of dissipativity for which the hyperbolic theory is well posed, but we introduce a new class of well-posed hyperbolic boundary problems: that of weakly dissipative boundary conditions. We then show that our system belongs to this class and is therefore well posed.
Among 3-dimensional sets of given Newtonian capacity, which shape minimizes the q-th moment (q>0) of electrostatic equilibrium measure? One readily shows it is the ball. But what if the set is confined to the plane? A centered disk is then the natural minimizer, yet the proof is quite different and involves a cylindrical variant of Baernstein’s star-function. The approach succeeds when 0 <q <= 2. Higher moments (q>2) remain a tantalizing open problem, as do the analogous questions for Riesz equilibrium measures.
Note: this talk does not assume any previous knowledge about capacities.
(Joint work with Carrie Clark, Univ. of Illinois Urbana–Champaign.)
The topic of the talk is existence of weak solutions to the pressureless Navier-Stokes system with nonlocal attraction--repulsion forces. We construct the solutions on the whole three-dimensional space, assuming that the viscosity coefficients are density-dependent. For the nonlocal term it is further assumed that the interaction kernel has the quadratic growth at infinity and almost quadratic singularity at zero. The main point of the construction is the derivation of the analogs of the Bresch--Desjardins and Mellet--Vasseur estimates in the nonlocal setting.
The motion of a compressible viscous barotropic fluid is described by the Navier-Stokes system. It is a system of hyperbolic-parabolic mixed-type PDEs. In this talk, we will study the so-called density patch problem: If we are given a density that is initially discontinuous across a C^(1+\alpha) curve alpha and alpha- Hölder continuous on the two disjoint components delimited by gamma, is this structure preserved in time?
An important quantity in the mathematical analysis of this system is the so-called effective flux, which was discovered by Hoff and Smoller in 1985. More precisely, the mathematical properties of this quantity play a crucial role in the study of the propagation of oscillations in compressible fluids (Serre, 1991), in the construction of weak solutions (P-L Lions 1996) or the propagation of discontinuity surfaces (Hoff 2002), to cite just a few examples. In the case of density-dependent viscosities, the behavior of the effective flux degenerates, which renders the analysis more subtle.
In the first part of this talk we will give an overview of our past and present research activity, highlighting the different fields of applied mathematics that have been considered so far. In the second part of the talk, I present a novel Finite Volume (FV) scheme on unstructured polygonal meshes that is provably compliant with the Second Law of Thermodynamics and the Geometric Conservation Law (GCL) at the same time. The governing equations are provided by a subset of the class of symmetric and hyperbolic thermodynamically compatible (SHTC) models introduced by Godunov in 1961. Specifically, our numerical method discretizes the equations for the conser- vation of momentum, total energy, distortion tensor and thermal impulse vector, hence accounting in one single unified mathematical formalism for a wide range of physical phenomena in continuum mechanics, spanning from ideal and viscous fluids to hyperelastic solids. By means of two conservative corrections directly embedded in the definition of the numerical fluxes, the new schemes are proven to satisfy two extra conservation laws, namely an entropy balance law and a geometric equation that links the distortion tensor to the density evolution. As such, the classical mass conservation equation can be discarded. Firstly, the GCL is derived at the continuous level, and subsequently it is satisfied by introducing the new concepts of general potential and generalized Gibbs relation. The new potential is nothing but the determinant of the distortion tensor, and the associated Gibbs relation is derived by introducing a set of dual or thermodynamic variables such that the GCL is retrieved by dot multiplying the original system with the new dual variables. Once compatibility of the GCL is ensured, thermodynamic compatibility is tackled in the same manner, thus achieving the satisfaction of a local cell entropy inequality. The two corrections are orthogonal, meaning that they can coexist simultaneously without interfering with each other. The compatibility of the new FV schemes holds true at the semi-discrete level, and time integration of the governing PDE is carried out relying on Runge-Kutta schemes. A large suite of test cases demonstrates the structure preserving properties of the schemes at the discrete level as well.
Wave breaking is a challenging subject that is not encompassed in the usual mathematical description of water waves. This is the consequence of the impossibility to represent the water-air interface as the graph of a function. In the first part of this presentation, we shall exhibit the strong non-linear nature of the breaking phenomena through the mathematical study of two water waves models: (1) KdV, whose solutions do not break and (2) Camassa-Holm, whose non-global solutions do break at some point. Next we shall discuss the way to incorporate multi-valued interfaces in the usual water waves problem before discussing whether or not this model efficiently describes the breaking phenomenon. In a third, ultimate, part we will present new numerical results that have been obtained using a finite-element code to solve the free-surface Navier-Stokes equations for an initial condition leading to a plunging jet. We compare these results with those obtained solving the Euler equation in the exact same configuration and conclude about the convergence of the two methods whenever the Reynolds number increases. We also discuss the influence of viscous dissipation on the overall shape of the wave.
L'imagerie haute résolution de la Terre, et en particulier de la croûte, est fondamentale pour la transition énergétique: pour la massiffication du stokage de CO2, une technologie mise en avant par le GIEC pour lutter contre le réchauffement climatique, mais aussi pour l'exploitation des ressources nécessaires pour la construction des infrastructures énergétiques éoliennes et salaires, et les batteries électriques. L'état de l'art pour l'imagerie haute résolution de la croûte repose sur une méthode appelée "inversion des formes d'ondes complètes". D'un point de vue mathématique, ceci revient à un problème d'estimation de paramètres d'une équation aux dérivées partielles (EDP) modélisant la propagation d'ondes dans le sous-sol à partir de données collectées ponctuellement en surface. Dans cette présentation, on introduit les bases géophysiques et mathématiques autour de cette méthode, avant de passer en revue des travaux de recherche menés au sein du projet SEISCOPE. Ces travaux recoupent les thèmes suivants: méthode d'optimisation de second-ordre basées sur des méthodes adjointes d'ordre deux, utilisation de distances transport optimal pour lutter contre le caractère mal posé du problème inverse, modélisation numérique 3D de la propagation d'onde dans l'approximation élastique, théorie des milieux équivalents (homogénisation) pour la propagation d'ondes en milieux élastiques, estimation des incertitudes pour les problèmes inverse de grande taille en se basant sur une méthode de filtre de Kalman d'ensemble.
Après un rappel sur les équations d'Euler 2D, je parlerai des tourbillons concentrés. J’exposerai les arguments principaux pour montrer la persistance de la concentration vers des points vérifiant le système des points vortex. Dans la seconde partie, je présenterai les équations des lacs qui peuvent se voir comme une généralisation d’Euler 3D axisymétrique sans swirl. Je montrerai que les points vortex se déplacent selon une loi de type « courbure binormale ». Ce travail est en collaboration avec Lars Eric Hientzsch et Evelyne Miot.
Geophysical phenomenon such as magnetic field reversal are a challenge to observe numerically. They are quite demanding in terms of numerical resources, and with the upcoming generation of exascale computers, it becomes necessary to ensure an efficient and full usage of such clusters. In geophysical and astrophysical flows, the classical method for parallelism is data parallelism. The physical domain is distributed across a large number of cores. The scaling of such a distribution can quickly saturate when the number of cores increases. Introducing an additional pipeline parallelism, through a distribution of a time interval across a number of cores, is a potential solution to use a larger number of cores, and perform numerical simulations of magnetic field reversals. When used by parallel in time schemes, time-steppers need to validate a few of criteria. We extracted 18 time-steppers from literature, from second to eighth order of accuracy. We will compare the accuracy and efficiency of those time steppers in the context of liquid planetary cores, in order to identify potential candidates to build parallel in time schemes.