A result of convergence for a mono-dimensional two-velocities lattice Boltzmann scheme

We consider a mono-dimensional two-velocities scheme used to approximate the solutions of a scalar hyperbolic conservative partial differential equation. We prove the convergence of the discrete solution toward the unique entropy solution by first estimating the supremum norm and the total variation of the discrete solution, and second by constructing a discrete kinetic entropy-entropy flux pair being given a continuous entropy-entropy flux pair of the hyperbolic system. We finally illustrate our results with numerical simulations of the advection equation and the Burgers equation

Lattice Boltzmann schemes with relative velocities

In this contribution, a new class of lattice Boltzmann schemes is introduced and studied. These schemes are presented in a framework that generalizes the multiple relaxation times method of d'Humi\`eres. They extend also the Geier's cascaded method. The relaxation phase takes place in a moving frame involving a set of moments depending on a given relative velocity field. We establish with the Taylor expansion method that the equivalent partial differential equations are identical to the ones obtained with the multiple relaxation times method up to the second order accuracy. The method is then performed to derive the equivalent equations up to third order accuracy

On the stability of a relative velocity lattice Boltzmann scheme for compressible Navier-Stokes equations

This paper studies the stability properties of a two dimensional relative velocity scheme for the Navier-Stokes equations. This scheme inspired by the cascaded scheme has the particularity to relax in a frame moving with a velocity field function of space and time. Its stability is studied first in a linear context then on the non linear test case of the Kelvin-Helmholtz instability. The link with the choice of the moments is put in evidence. The set of moments of the cascaded scheme improves the stability of the d'Humi\`eres scheme for small viscosities. On the contrary, a relative velocity scheme with the usual set of moments deteriorates the stability

Shape minimization of the dissipated energy in dyadic trees

In this paper, we study the role of boundary conditions on the optimal shape of a dyadic tree in which flows a Newtonian fluid. Our optimization problem consists in finding the shape of the tree that minimizes the viscous energy dissipated by the fluid with a constrained volume, under the assumption that the total flow of the fluid is conserved throughout the structure. These hypotheses model situations where a fluid is transported from a source towards a 3D domain into which the transport network also spans. Such situations could be encountered in organs like for instance the lungs and the vascular networks. Two fluid regimes are studied: (i) low flow regime (Poiseuille) in trees with an arbitrary number of generations using a matricial approach and (ii) non linear flow regime (Navier-Stokes, moderate regime with a Reynolds number 100) in trees of two generations using shape derivatives in an augmented Lagrangian algorithm coupled with a 2D/3D finite elements code to solve Navier-Stokes equations. It relies on the study of a finite dimensional optimization problem in the case (i) and on a standard shape optimization problem in the case (ii). We show that the behaviours of both regimes are very similar and that the optimal shape is highly dependent on the boundary conditions of the fluid applied at the leaves of the tree.Comment: \`a para\^itre dans Discrete Contin. Dyn. Syst. (B

Curious convergence properties of lattice Boltzmann schemes for diffusion with acoustic scaling

We consider the D1Q3 lattice Boltzmann scheme with an acoustic scale for the simulation of diffusive processes. When the mesh is refined while holding the diffusivity constant, we first obtain asymptotic convergence. When the mesh size tends to zero, however, this convergence breaks down in a curious fashion, and we observe qualitative discrepancies from analytical solutions of the heat equation. In this work, a new asymptotic analysis is derived to explain this phenomenon using the Taylor expansion method, and a partial differential equation of acoustic type is obtained in the asymptotic limit. We show that the error between the D1Q3 numerical solution and a finite-difference approximation of this acoustic-type partial differential equation tends to zero in the asymptotic limit. In addition, a wave vector analysis of this asymptotic regime demonstrates that the dispersion equation has nontrivial complex eigenvalues, a sign of underlying propagation phenomena, and a portent of the unusual convergence properties mentioned above

Isotropy conditions for lattice Boltzmann schemes. Application to D2Q9

In this paper, we recall the linear version of the lattice Boltzmann schemes in the framework proposed by d'Humi\'eres. According to the equivalent equations we introduce a definition for a scheme to be isotropic at some order. This definition is chosen such that the equivalent equations are preserved by orthogonal transformations of the frame. The property of isotropy can be read through a group operation and then implies a sequence of relations on relaxation times and equilibrium states that characterizes a lattice Boltzmann scheme. We propose a method to select the parameters of the scheme according to the desired order of isotropy. Applying it to the D2Q9 scheme yields the classical constraints for the first and second orders and some non classical for the third and fourth orders

A stability property for a mono-dimensional three velocities scheme with relative velocity

In this contribution, we study a stability notion for a fundamental linear one-dimensional lattice Boltzmann scheme, this notion being related to the maximum principle. We seek to characterize the parameters of the scheme that guarantee the preservation of the non-negativity of the particle distribution functions. In the context of the relative velocity schemes, we derive necessary and sufficient conditions for the non-negativity preserving property. These conditions are then expressed in a simple way when the relative velocity is reduced to zero. For the general case, we propose some simple necessary conditions on the relaxation parameters and we put in evidence numerically the non-negativity preserving regions. Numerical experiments show finally that no oscillations occur for the propagation of a non-smooth profile if the non-negativity preserving property is satisfied

L'évaluabilité du Crédit d'Impôt pour la Transition Énergétique

Le Crédit d’Impôt pour la Transition Energétique (CITE), anciennement Crédit d’Impôt Développement Durable (CIDD), est un dispositif fiscal dont l’objectif depuis 2005 est d’inciter les ménages à réaliser des travaux de performance énergétique dans leur logement. La rénovation énergétique du parc de logements français constitue un défi majeur pour les pouvoirs publics et s’inscrit dans l’ambition française d’économies d’énergie et de réduction des émissions de gaz à effet de serre (GES). Au cours des dix dernières années, le dispositif a subi de nombreux ajustements, avant de devenir en 2014 le CITE, avec un taux unique à 30%. Dans un contexte de rationalisation des dépenses publiques il apparaît nécessaire d’interroger et d’expliciter les ressorts d’un tel outil de politique publique. Des évaluations, souvent ponctuelles et centralisées, ont été menées au cours de son exercice pour tester son efficacité énergétique et son efficience conomique avec des résultats contrastés fondés sur des modèles théoriques ou des enquêtes déclaratives. Cette étude porte sur les pistes d’évaluabilité du CITE, dans l’objectif de contribuer à son amélioratio