The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always correctly...

The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always...

We propose and analyse two convergent fully discrete schemes to solve the incompressible Navier-Stokes-Nernst-Planck-Poisson system. The first scheme converges to weak solutions satisfying an energy and an entropy dissipation law. The second scheme uses Chorin's projection method to obtain an efficient approximation that converges to strong solutions at optimal rates.

The incompressible MHD equations couple Navier-Stokes equations with Maxwell's equations to describe the flow of a viscous, incompressible, and electrically conducting fluid in a Lipschitz domain $\Omega \subset {ℝ}^3$. We verify convergence of iterates of different coupling and decoupling fully discrete schemes towards weak solutions for vanishing discretization parameters. Optimal first order of convergence is shown in the presence of strong solutions for a splitting scheme which decouples the computation of velocity...

We introduce a new condition which extends the definition of sticky particle dynamics to the case of discontinuous initial velocities $u_0$ with negative jumps. We show the existence of a stochastic process and a forward flow $\phi$ satisfying $X_{s+t}=\phi (X_s,t,P_s,u_s)$ and $\mathrm{d}{X}_t=\mathrm{E}[u_0\left(X_0\right)/X_t]\mathrm{d}t$, where $P_s=P{X}_s^{-1}$ is the law of $X_s$ and $u_s\left(x\right)=\mathrm{E}[u_0\left(X_0\right)/X_s=x]$ is the velocity of particle $x$ at time $s\ge 0$. Results on the flow characterization and Lipschitz continuity are also given.Moreover, the map $(x,t)\mapsto M(x,t):=P(X_t\le x)$ is the entropy solution of a scalar conservation law ${\partial }_{t}M+{\partial }_x\left(A\left(M\right)\right)=0$ where the flux $A$ represents the particles...

Discrete-velocity approximations represent a popular way for computing the Boltzmann collision operator. The direct numerical evaluation of such methods involve a prohibitive cost, typically O(N2d + 1) where d is the dimension of the velocity space. In this paper, following the ideas introduced in [C. Mouhot and L. Pareschi, C. R. Acad. Sci. Paris Sér. I Math. 339 (2004) 71–76, C. Mouhot and L. Pareschi, Math. Comput. 75 (2006) 1833–1852], we derive fast summation techniques for the evaluation of...

In order to handle the flow of a viscous incompressible fluid in a porous medium with cracks, the thickness of which cannot be neglected, we consider a model which couples the Darcy equations in the medium with the Stokes equations in the cracks by a new boundary condition at the interface, namely the continuity of the pressure. We prove that this model admits a unique solution and propose a mixed formulation of it. Relying on this formulation, we describe a finite element discretization and derive...

In order to handle the flow of a viscous incompressible fluid in a porous medium with cracks, the thickness of which cannot be neglected, we consider a model which couples the Darcy equations in the medium with the Stokes equations in the cracks by a new boundary condition at the interface, namely the continuity of the pressure. We prove that this model admits a unique solution and propose a mixed formulation of it. Relying on this formulation, we describe a finite element discretization and derive...

We consider the propagation of internal waves at the interface between two layers of immiscrible fluids of different densities, under the rigid lid assumption, with the presence of surface tension and with uneven bottoms. We are interested in the case where the flow has a Boussinesq structure in both the upper and lower fluid domains. Following the global strategy introduced recently by Bona, Lannes and Saut [J. Math. Pures Appl. 89 (2008)], we derive an asymptotic model in this regime, namely the...

This work studies the heat equation in a two-phase material with spherical inclusions. Under some appropriate scaling on the size, volume fraction and heat capacity of the inclusions, we derive a coupled system of partial differential equations governing the evolution of the temperature of each phase at a macroscopic level of description. The coupling terms describing the exchange of heat between the phases are obtained by using homogenization techniques originating from [D. Cioranescu, F. Murat,...

In this work, we are interested in two different diffusion models for multicomponent mixtures. We numerically recover experimental results underlining the inadequacy of the usual Fick diffusion model, and the importance of using the Maxwell-Stefan model in various situations. This model nonlinearly couples the mole fractions and the fluxes of each component of the mixture. We then consider a subregion of the lower part of the lung, in which we compare...

Modeling the kinetics of a precipitation dissolution reaction occurring in a porous medium where diffusion also takes place leads to a system of two parabolic equations and one ordinary differential equation coupled with a stiff reaction term. This system is discretized by a finite volume scheme which is suitable for the approximation of the discontinuous reaction term of unknown sign. Discrete solutions are shown to exist and converge towards a weak solution of the continuous problem. Uniqueness...

This contribution deals with the numerical simulation of dislocation dynamics. Dislocations are described by means of the evolution of a family of closed or open smooth curves $\Gamma \left(t\right):S\to {ℝ}^2$, $t\geqq 0$. The curves are driven by the normal velocity $v$ which is the function of curvature $\kappa$ and the position. The evolution law reads as: $v=-\kappa +F$. The motion law is treated using direct approach numerically solved by two schemes, i. e., backward Euler semi-implicit and semi-discrete method of lines. Numerical stability is improved...