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International audience; In this article, we consider the time dependent convection-diffusion-reaction equation coupled with the Darcy equation. We propose a numerical scheme based on finite element methods for the discretization in space and the implicit Euler method for the discretization in time. We establish optimal a posteriori error estimates with two types of computable error indicators, the first one linked to the time discretization and the second one to the space discretization. Finally, numerical investigations are performed and presented.

FVCA 2017 : International Conference on Finite Volumes for Complex Applications; This benchmark proposes test-cases to assess innovative finite volume type methods developped to solve the equations of incompressible fluid mechanics. Emphasis is set on the ability to handle very general meshes, on accuracy, robustness and computational complexity. Two-dimensional as well as three-dimensional tests with known analytical solutions are proposed for the steady Stokes and both steady and unsteady Navier-Stokes equations, as well as classical lid-driven cavity tests.

We present an application of the discrete duality finite volume method to the numerical approximation of the vorticity-velocity-pressure formulation of the two-dimensional Stokes equations, associated to various nonstandard boundary conditions. The finite volume method is based on the use of discrete differential operators obeying some discrete duality principles. The scheme may be seen as an extension of the classical Marker and Cell scheme to almost arbitrary meshes, thanks to an appropriate choice of degrees of freedom. The efficiency of the scheme is illustrated by numerical examples over unstructured triangular and locally refined nonconforming meshes, which confirm the theoretical convergence analysis led in the article. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1–30, 2015

By studying the structure of the discrete kernel of the linear acoustic operator discretized with a Godunov scheme, we clearly explain why the behaviour of the Godunov scheme applied to the linear wave equation deeply depends on the space dimension and, especially, on the type of mesh. This approach allows us to explain why, in the periodic case, the Godunov scheme applied to the resolution of the compressible Euler or Navier-Stokes system is accurate at low Mach number when the mesh is triangular or tetrahedral and is not accurate when the mesh is a 2D (or 3D) cartesian mesh. This approach confirms also the fact that a Godunov scheme remains accurate when it is modified by simply centering the discretization of the pressure gradient.

International audience; An efficient and fully computable a posteriori error bound is derived for the discrete duality finite volume discretization of the Laplace equation on very general twodimensional meshes. The main ingredients are the equivalence of this method with a finite element like scheme and tools from the finite element framework. Numerical tests are performed with a stiff solution on highly nonconforming locally refined meshes and with a singular solution on triangular meshes.

A new finite volume method is presented for discretizing the two-dimensional Maxwell equations. This method may be seen as an extension of the covolume type methods to arbitrary, possibly non-conforming or even non-convex, n-sided polygonal meshes, thanks to an appropriate choice of degrees of freedom. An equivalent formulation of the scheme is given in terms of discrete differential operators obeying discrete duality principles. The main properties of the scheme are its energy conservation, its stability under a CFL-like condition, and the fact that it preserves Gauss' law and divergence free magnetic fields. Second-order convergence is demonstrated numerically on non-conforming and distorted meshes.

We compare two models used to compute the internal hydrodynamics of a gas centrifuge. The scoop action is taken into account through boundary conditions on the flow entering the bowl of the centrifuge in the first model, and through sinks and drag forces in the chambers of the centrifuge in the second. The numerical approximations of the models are based on a finite volume scheme on staggered rectangular grids and on a fixed-point iterative method. Convergence of the approximations is studied numerically on a family of refined grids and comparisons of the two models are discussed for the Iguacu centrifuge. It appears that linear computations on rough grids are sufficient in the first model to correctly predict the separative power of the centrifuge, while other parameters like the return flow or the drag forces require finer meshes and non-linear computations in the second model.

We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green formula. This method generalizes an existing finite volume method that requires "Voronoi-type" meshes. We show the equivalence of this finite volume method with a non-conforming finite element method with basis functions being P 1 on the cells, generally called "diamond-cells", of a third mesh. Under geometrical conditions on these diamond- cells, we prove a first-order convergence both in the H 1 norm and in the L 2 norm. Superconvergence results are obtained on certain types of homothetically refined grids. Finally, numerical experiments confirm these results and also show second-order convergence in the L 2 norm on general grids. They also indicate that this method performs particularly well for the approximation of the gradient of the solution, and may be used on degenerating triangular grids. An example of application on non- conforming locally refined grids is given.

This paper deals with the linear response of a plasma in a one-dimensional bounded geometry under the action of a time-periodic electric field. The nonlinear Vlasov equation is solved by following the characteristic curves until they reach the boundary of the domain, where the distribution function of the incoming particles is supposed to be known and independent of time. Then, a first-order Taylor expansion in the velocity variable is performed, thanks to an approximation of the exact characteristics by the unperturbed ones. The resulting first-order correction to the distribution function is finally integrated over velocities to yield the dielectric function. The special case of a plane wave for the electric field is examined and the results are compared with the more usual unbounded case: the integral does not present any singularity in the vicinity of resonant particles and the dielectric function depends on the distance to the boundary and tends to the usual infinite-geometry value when this distance tends to infinity, with a rate of convergence proportional to its inverse square root. Numerical examples are provided for illustration.

We present an extension of the Optimized Schwarz Waveform Relaxation method with Robin transmission conditions to finite volume schemes of DDFV type (Discrete Duality Finite Volumes) for solving heterogeneous time-dependent advection-diffusion problems. We propose a new strategy which is well adapted to domain decomposition for coupling upwind discretization of the convection with diffusion in the context of a finite volume method. The method is proven to be well posed and we prove the convergence of the iterative algorithm. Then we present numerical results to illustrate the method.

A self-consistent nonlinear model of an isotope separation process based on selective ion cyclotron resonance heating in a magnetized plasma is presented, and its numerical resolution is described. The response of the electrons to the electromagnetic field is modeled by a cold and linear conductivity tensor, while a particle method is used to solve nonlinear Vlasov equations for the ions. The resolution of the time-harmonic Maxwell equations is achieved by a finite-element method. Both steps are coupled by an iterative procedure, which shows fast convergence. Results are presented for the case of a solenoidal launching antenna.

Usually, non-stationary numerical calculations in electromagnetics are based on the hyperbolic evolution equations for the electric and magnetic fields and leave Gauss' law out of consideration because the latter is a consequence of the former and of the charge conservation equation in the continuous case. However, in the simulation of the self-consistent movement of charged particles in electromagnetic fields, it is a well-known fact that the approximation of the particle motion introduces numerical errors and that, consequently, the charge conservation equation is not satisfied on the dicrete level. Then, in order to avoid the increase of errors in Gauss' law, a divergence cleaning step which solves a Poisson equation for a correction potential is often added. In the present paper, a new method for incorporating Gauss' law into non-stationary electromagnetic simulation codes is developed, starting from a constrained formulation of the Maxwell equations. The resulting system is hyperbolic, and the divergence errors propagate with the speed of light to the boundary of the computational domain. Furthermore, the basic ideas of the numerical approximation are introduced and the extended hyperbolic system is treated numerically within the framework of high-resolution finite-volume schemes. Simulation results obtained with this new technique for pure electromagnetic wave propagation and for an electromagnetic particle-in-cell computation are presented and compared with other methods.

Far field simulations of underground nuclear waste disposal involve a number of challenges for numerical simulations: widely differing lengths and time-scales, highly variable coefficients and stringent accuracy requirements. In the site under consideration by the French Agency for NuclearWaste Management (ANDRA), the repository would be located in a highly impermeable geological layer, whereas the layers just above and below have very different physical properties (see [1]).

We investigate the accuracy of the Godunov scheme applied to the variable cross-section acoustic equations. Contrarily to the constant cross-section case, the accuracy issue of this scheme in the low Mach number regime appears even in the one-dimensional case; on the other hand, we show that it is possible to construct another Godunov type scheme which is accurate in the low Mach number regime.

Pulsed-power diodes have been developed at the Forschungszentrum Karlsruhe and are the objects of extensive experimental as well as numerical investigations. The electrical behavior of the diodes is substantially influenced by a charged particle flow forming a non-neutral plasma inside these devices. A detailed understanding of the fundamental time-dependent phenomena (e.g., the origin of instabilities) caused by this plasma requires the solution of the Maxwell-Lorentz equations for realistic configurations with a very accurate replica of the border of the domain, where several kinds of boundary conditions are imposed. An attractive method to attack this non-linear equations numerically is the particle-in-cell (PIC) technique. As a preliminary to use the PIC approach, the relevant diode domain has to be covered by an appropriate computational mesh. Therefore, we adopt a grid model based on boundary-fitted coordinates resulting in a quadrilateral mesh zone arrangement with regular data structure. The numerical solution of the Maxwell equations in time domain is obtained by using a finite-volume (FV) approach on a non-rectangular quadrilateral mesh in two space dimensions. A very favorable property of these modern FV schemes consists in the fact that they combine inherent robustness at steep gradients with accurate resolution. In the context of self-consistent charged particle simulation in electromagnetic fields the coupling of a high-resolution FV Maxwell solver with the PIC method is a new way of approximation.

The problem of spurious solutions due to the violation of Gauss’ law in computational electromagnetics is avoided by solving an equivalent Maxwell system that takes this constraint into account. A second-order accurate finite-volume method is proposed to solve this linear, first-order strictly hyperbolic reformulated system. Numerical examples demonstrate the validity of this approach.

We present a high-resolution finite-volume Godunov-type Maxwell solver for three-dimensional unstructured meshes, based on the purely hyperbolic Maxwell (PHM) system, which is established by introducing two additional degrees of freedom into the evolutionary part of the Maxwell equations and coupling them with the elliptical constraints given by Gaus’ law and the ∇ · B = 0 statement. This model allows for possible errors in the charge conservation equation as may occur in particle-in-cell simulations, and yields approximative solutions of the conventional Maxwell equations. Numerical results demonstrate the relevance of the correction approach when the charge conservation equation is violated.

International audience; This article deals with the modelling and formulation of compositional gasliquid Darcy flow. Our model includes an advanced boundary condition at the interface between the porous medium and the atmosphere accounting for convective mass and energy transfer, liquid evaporation, and liquid outflow. The formulation is based on a fixed set of unknowns whatever the set of present phases. The thermodynamical equilibrium is expressed as complementary constraints. The model and its formulation are applied to the simulation of the Bouillante high energy geothermal field in Guadeloupe characterized by a high temperature closed to the surface.

International audience; In steady-state regimes, water circulating in the nuclear power plants pipes behaves as a low Mach number flow. However, when steep phenomena occur, strong shock waves are produced. Herein, a fractional step approach allowing to decouple the convective from the acoustic effects is proposed. The originality is that the splitting between these two parts of the physics evolves dynamically in time according to the Mach number. The first one-dimensional explicit and implicit numerical results on a wide panel of Mach numbers show that this approach is as accurate and CPU-consuming as a state of the art Lagrange-Projection-type method.

International audience; Flow and transport in fracturedporous media are of paramount importance for many applications such as petroleum exploration and production, geological storage of carbon dioxide, hydrogeology, or geothermal energy. We consider here the two-phase discrete fracture model introduced in [3] which represents explicitly the fractures as codimension one surfaces immersed in the surrounding matrix domain. Then, the two-phase Darcy flow in the matrix is coupled with the two-phase Darcy flow in the fractures using transmission conditions accounting for fractures acting either as drains or barriers. The model takes into account complex networks of fractures, discontinuous capillary pressure curves at the matrix fracture interfaces and can be easily extended to account for gravity including in the width of the fractures. It also includes a layer of damaged rock at the matrix fracture interface with its own mobility and capillary pressure functions. In this work, the convergence analysis carried out in [3] in the framework of gradient discretizations [2] is extended to obtain the uniform-in-time convergence of the discrete solutions to a weak solution of the model.

International audience; A domain decomposition algorithm is proposed to couple at the interface a gas liquid compositional Darcy flow and a compositional free gas flow. At each time step, our algorithm solves iteratively the nonlinear system coupling the compositional Darcy flow in the porous medium, the RANS gas flow in the free flow domain, and the convection diffusion of the species in the free flow domain. In order to speed up the convergence of the algorithm, the transmission conditions at the interface are replaced by Robin boundary conditions. Each Robin coefficient is obtained from a diagonal approximation of the Dirichlet to Neumann operator related to a scalar simplified model in the neighbouring subdomain. The efficiency of our domain decomposition algorithm is assessed in the case of the modelling of the mass exchanges at the interface between the geological formation and the ventilation galleries of geological radioactive waste disposal.