Convergence Analysis on Unstructured Meshes of a DDFV Method for Flow Problems with Full Neumann Boundary Conditions

A Discrete Duality Finite Volume (DDFV) method to solve on unstructured meshes the flow problems in anisotropic nonhomogeneous porous media with full Neumann boundary conditions is proposed in the present work. We start with the derivation of the discrete problem. A result of existence and uniqueness of a solution for that problem is given thanks to the properties of its associated matrix combined with adequate assumptions on data. Their theoretical properties, namely, stability and error estimates (in discrete energy norms and L2-norm), are investigated. Numerical test is provided

Sediment transport models in Generalized shear shallow water flow equations

International audienceThe classical sediment transport models based on shallow water equations (SWE) describes the hydro-morphodynamic process without horizontal velocity shear along the vertical and considers that the fluid velocity is equal to sediment velocity. The classical shear shallow water (SSW) with friction and topography source terms assumes that the fluid density is uniform in all the space. Nevertheless, for the coastal flows with sediment transport we are interested in it seems essential to consider these shear effects, the phase lag effect and the nonhomogenous ness of fluid density. In this paper, we develop new sediment transport models incorporating the shear velocity along the vertical, the phase lag effect and the spatial variation of the fluid density. The starting point is the 2D equations for the evolution of mixing quantities and sediment volume rate. These equations describe the motion of fluid mixing in a domain bounded by a dynamical water surface and water bed. Contrary to the classical sediment transport models, the second-order vertical fluctuations of the horizontal velocity are considered. Considering the kinematic conditions on the moving surfaces, we apply an average along the depth on the three-dimensional equations to obtain simplified equations. The resulting model has a wider range of validity and integrates the morphodynamic processes proposed in the literature. The proposed mathematical derivation is in the context of recent developments with the additional presence of sediment and a dynamic bed

Expression en termes d'énergie pour la perméabilité absolue effective. Application au calcul numérique d'écoulements diphasiques en milieu poreux

Le cadre de ce travail est le calcul des paramètres pétrophysiques effectifs d'un milieu poreux hétérogène pour le simulateur de réservoirs pétroliers. Après le choix d'un modèle d'écoulement dans un milieu poreux hétérogène comportant une microstructure périodique nous rappelons brièvement les grandes étapes de la méthode des échelles multiples pour l'homogénéisation de ce modèle. Cela nous conduit à la formule classique d'homogénéisation de la perméabilité absolue. Par la suite nous présentons une démarche originale permettant de passer de cette formule classique à une formule plus simple (d'un point de vue numérique) s'exprimant en termes d'énergie dissipée par les forces de viscosité locales et caractérisant le milieu hétérogène périodique considéré. Nous démontrons ensuite, sous certaines hypothèses, l'égalité entre les énergies dissipées par les forces de viscosité associées respectivement à l'écoulement local et à l'écoulement macroscopique. Nous terminons par la présentation de quelques résultats numériques concernant des modèles d'écoulement diphasique incompressible

Finite volume methods for one-dimensional flow problems with discontinuous coefficients: A new approach based up on second order convergence

The finite volume methods are well known as powerful tools to address system of conservation equations. Over the past two decades a lot of effort has been made to put in place a mathematical framework for the theoretical analysis of Finite Volume Methods for second order differential operators. A perfect illustration of this progress is the development of Discrete Duality Finite Volume Methods (DDFV, for short). Following that trend we expose in this work a new approach of Finite Volume Schemes for second order elliptic problems in one dimension space, involving discontinuities in the diffusion coefficients. That new approach is based up on a system of two grids: a primary grid for having a control on localization of discontinuity points of the diffusion coefficient and a control-volume grid (which is not the dual of the primary grid) defined in such a way to get a second order approximation of the fluxes. The algebraic structure of the discrete problem we have got shows that this new approach is not a 1-D version of the DDFV even that both of them are cell-centered and vertex-centered. The proposed scheme leads to a symmetric, positive definite algebraic system and its solution meets the maximum principle. We have shown the second order convergence of the proposed scheme for pure diffusion problems on any primary grid (respecting quasi-uniformity of grid blocs). The second order convergence still holds for diffusion-reaction problems if the primary mesh elements are uniform, even if the diffusion coefficient gets discontinuity points. The ongoing work on the extension of the proposed method to two-dimensional problems is promising in terms of avoiding the computation of the equivalent diffusion coefficient to allocate to the diamond mesh elements as required from application of DDFV

Finite volume methods for one-dimensional flow problems with discontinuous coefficients: A new approach based up on second order convergence

The finite volume methods are well known as powerful tools to address system of conservation equations. Over the past two decades a lot of effort has been made to put in place a mathematical framework for the theoretical analysis of Finite Volume Methods for second order differential operators. A perfect illustration of this progress is the development of Discrete Duality Finite Volume Methods (DDFV, for short). Following that trend we expose in this work a new approach of Finite Volume Schemes for second order elliptic problems in one dimension space, involving discontinuities in the diffusion coefficients. That new approach is based up on a system of two grids: a primary grid for having a control on localization of discontinuity points of the diffusion coefficient and a control-volume grid (which is not the dual of the primary grid) defined in such a way to get a second order approximation of the fluxes. The algebraic structure of the discrete problem we have got shows that this new approach is not a 1-D version of the DDFV even that both of them are cell-centered and vertex-centered. The proposed scheme leads to a symmetric, positive definite algebraic system and its solution meets the maximum principle. We have shown the second order convergence of the proposed scheme for pure diffusion problems on any primary grid (respecting quasi-uniformity of grid blocs). The second order convergence still holds for diffusion-reaction problems if the primary mesh elements are uniform, even if the diffusion coefficient gets discontinuity points. The ongoing work on the extension of the proposed method to two-dimensional problems is promising in terms of avoiding the computation of the equivalent diffusion coefficient to allocate to the diamond mesh elements as required from application of DDFV

A finite volume approximation for second order elliptic problems with a full matrix on quadrilateral grids: derivation of the scheme and a theoretical analysis

International audienceWe present in this paper a finite-volume based flexible Multi-Point Flux Approximation method (MPFA method, in short) displaying strong capabilities to handle flow problems in non-homogeneous anisotropic media. When the diffusion coefficient governing the flow is a full matrix with constant components, the discrete system is symmetric positive definite even if this matrix is only positive definite. In addition, if the diffusion coefficient is diagonal, the discrete system is reduced to two independent discrete systems corresponding to well-known cell-centered and vertex-centered finite volume methods. A stability result and error estimates are given in L 2 − and L ∞ −norm and in a discrete energy norm as well. These results have been confirmed by numerical experiments. In this connection, a comparison of our MPFA method with the MPFA O-method has been performed. A finite volume approximation for second order elliptic problems with a full matrix on quadrilateral grids: derivation of the scheme and a theoretical analysi

Convergence analysis of an MPFA method for flow problems in anisotropic heterogeneous porous media

International audienceOur purpose in this paper is to present the theoretical analysis of a Multi-Point Flux Approximation method (MPFA method). We start with the derivation of the discrete problem, and then we give a result of existence and uniqueness of a solution for that problem. As in finite element theory, Lagrange interpolation is used to define three classes of continuous and locally polynomial approximate solutions. For analyzing the convergence of these different classes of solutions, the notions of weak and weak-star MPFA approximate solutions are introduced. Their theoretical properties, namely stability and error estimates (in discrete energy norms, L 2 − norm and L ∞ − norm), are investigated. These properties play a key role in the analysis (in terms of error estimates for diverse norms) of different classes of continuous and locally polynomial approximate solutions mentioned above

Solution volumes finis polynomiale par morceaux pour les problèmes de Diffusion-Convection admettant des solutions continues

Nous présentons dans ce papier un concept de solution volumes finis continue pour des problèmes de diffusion-convection avec des données régulières. Nous comparons dans certains cas particuliers la solution proposée avec la solution volumes finis classiques (qui est une fonction constante par morceaux) et la solution exacte. La norme L2 de l'erreur est déduite des simulations numériques. Journal des Sciences Pour l'Ingénieur Vol. 6, 2006: 56-6