Search

he present paper is focused on the proof of the convergence of the discrete implicit Marker-andCell (MAC) scheme for time-dependent Navier–Stokes equations with variable density and variableviscosity. The problem is completed with homogeneous Dirichlet boundary conditions and is discretized according to a non-uniform Cartesian grid. A priori-estimates on the unknowns are obtained,and along with a topological degree argument they lead to the existence of a solution of the discretescheme at each time step. We conclude with the proof of the convergence of the scheme toward thecontinuous problem as mesh size and time step tend toward zero with the limit of the sequence ofdiscrete solutions being a solution to the weak formulation of the problem. Finite volume methods;MAC scheme; incompressible Navier–Stokes equations; variable density and viscosity; transportequations.

The present paper is focused on the proof of the convergence of the discrete implicit Marker-and-Cell (MAC) scheme for time-dependent Navier--Stokes equations with variable density and variable viscosity. The problem is completed with homogeneous Dirichlet boundary conditions and is discretized according to a non-uniform Cartesian grid. A priori-estimates on the unknowns are obtained, and along with a topological degree argument they lead to the existence of a solution of the discrete scheme at each time step. We conclude with the proof of the convergence of the scheme toward the continuous problem as mesh size and time step tend toward zero with the limit of the sequence of discrete solutions being a solution to the weak formulation of the problem.

International audience

Another similar lecture was made at the Séminaire d'Analyse Appliquée du LAMFA, Mai 2009, Université de Picardie Jules Verne, Amiens.; International audience

National audience

The same lecture was made at the 1st Workshop on Solution Methodologies for Direct and Inverse Scattering Problems, INRIA Magique3D, Université de Pau et des Pays de l'Adour, December 12-14 2007.; International audience

Emory University; International audience

International audience

International audience

International audience

Dynamics of viscous shocks is considered in the modular Burgers equation, where the time evolution becomes complicated due to singularities produced by the modular nonlinearity. We prove that the viscous shocks are asymptotically stable under odd and general perturbations. For the odd perturbations, the proof relies on the reduction of the modular Burgers equation to a linear diffusion equation on a half-line. For the general perturbations, the proof is developed by converting the time-evolution problem to a system of linear equations coupled with a nonlinear equation for the interface position. Exponential weights in space are imposed on the initial data of general perturbations in order to gain the asymptotic decay of perturbations in time. We give numerical illustrations of asymptotic stability of the viscous shocks under general perturbations., Comment: 35 pages; 2 figures

We introduce a strategy for the simulation a particulate flow in a 3-dimensional domain. The particles are assumed to be rigid, and the homogeneous fluid flow to be governed by the incompressible Navier–Stokes equations. The system is solved using a predictor-corrector scheme for the Navier–Stokes equations with variable density. The latter scheme is adapted to take into account the solid domain by adopting a volume penalization method. In order to advect efficiently the particles, the approximation of the mass balance equation is carried out by an anti-dissipative scheme similar to the Ultra-Bee scheme. We conclude with numerical tests in the context of particulate flows.

We are concerned with modelling a particulate flow in a three-dimensional domain. The particles are assumed to be rigid, allowing us to describe their motion using the Newton laws. As we aim to take into account complex shapes for the solid inclusions, we adopt volume penalization methods. Those methods allow us to extend the fluid problem inside the solid domain by assimilating the particle as a porous medium. The homogeneous fluid flow is governed by the incompressible Navier-Stokes equations. The whole problem is solved with a projection-correction method using finite volumes and a staggered mesh to ensure the inf-sup condition for the stability. Regarding the transport of the particles, a marker-based front tracking method is used for the fluid-solid interface, as well as a collision strategy. Both penalization methods are studied and compared in the context of particulate flows.

An efficient preconditioner is developed for solving the Helmholtz problem in both high and low frequency (wavenumber) regimes. The preconditioner is based on hierarchical unknowns on nested grids, known as incremental unknowns (IU). The motivation for the IU preconditioner is provided by an eigenvalue analysis of a simplified Helmholtz problem. The performance of our preconditioner is tested on the iterative solution of two-dimensional electromagnetic scattering problems. When compared with other well-known methods, our technique is shown to often provide a better numerical efficacy and, most importantly, to be more robust. Moreover, for the best performance, the number of IU levels used in the preconditioner should be designed for the coarsest grid to have roughly two points per linear wavelength. This result is consistent with the conventional sampling criteria for wave phenomena in contrast with existing IU applications for solving the Laplace/Poisson problem, where the coarsest grid comprises just one interior point. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

Concept of edge polarizability produced by support of the edge quantum states is demonstrated by the example of graphene nanoribbon with zigzag edge configuration. This concept allows to formulate physically correct edge conditions for electromagnetic field in nanostructures instead of ordinary Meixner conditions in classical electrodynamics.

The solution of the Poisson equation is an essential stage in the calculation of quantum properties of materials as it appears both in the electrostatic Hartree term and in the Fock exchange operator. We discuss here an integral method for the calculation of Poisson potential and demonstrate its integration with a Density Functional Theory (DFT) code. We show some results on elongated carbon chains.

The calculation of Fock-Exchange integral is becoming an important computational challenge in solving the quantum properties of systems with many electrons. In this work we show how techniques that are used in electromagnetic simulations for the similar Poisson equation can help in accelerating the calculation of such exchange integrals. The approach we present can be especially efficient for quasi-planar systems such as Graphene nano-ribbons (GNRs) or elongated systems such as carbon nano-tubes (CNTS) and the 1D Carbyne.

In an earlier paper (Poullet and Boag, 2007) [1], we developed an efficient incremental unknowns (IU) preconditioner for solving the two-dimensional (2D) Helmholtz problem in both high and low frequency (wavenumber) regimes. The multilevel preconditioning scheme involves separation of each grid into a coarser grid of the following level and a complementary grid on which the IUs are defined by interpolation. This approach is efficient as long as the mesh size of the coarsest grid is sufficiently small compared to the wavelength. In order to overcome this restriction, the authors introduced recently (in Poullet and Boag (2010) [2]) a modified IU method combining the conventional interpolation with the Helmholtz equation based interpolation (EBI). The EBI coefficients are derived numerically using a sufficiently large set of analytic solutions of the Helmholtz equation on a special hierarchy of stencils. The modified IUs using Helmholtz EBI are shown to provide improved preconditioning on the coarse scales where the conventional interpolation can not be employed. This study deals with the extension of this idea for solving the three-dimensional (3D) Helmholtz equation.

We deal with the time-dependent Navier-Stokes equations with Dirichlet boundary conditions on all the domain or, on a part of the domain and open boundary conditions on the other part. It is shown numerically that a staggered mesh with penalty-projection method yields reasonable good results for solving the above mentioned problem. Similarly to the results obtained recently by other scientists using finite element method (FEM) [1] and [2] (with the rotational pressure-correction method for the latter), we confirm that the penalty-projection scheme with spatial discretization of the Marker And Cell method (MAC) [3] is compatible with our problem.

We mainly study a system of two coupled nonlinear Schrödinger equations where one equation includes gain and the other one includes losses. This model constitutes a generalization of the model of pulse propagation in bire-fringent optical fibers. We answered partially to a question of some authors in [8], that in the Manakov case, the solution stays in L 2 (0, T ; H 1), that means that the solution can not blow up in finite time. More precisely, the bound that is provided in this paper does not seem to be optimal but different than those that has been given from a previous study [8]. Thanks to the way we treat the a priori estimate, we obtain a sharp bound in L 2 (0, T ; H 1), which would be difficult to reach from the study of other authors [8]. The result is illustrated by numerical results which have been obtained with a finite element solver well adapted for that purpose.