Entropy estimates for a class of schemes for the euler equations

In this paper, we derive entropy estimates for a class of schemes for the Euler equations which present the following features: they are based on the internal energy equation (eventually with a positive corrective term at the righ-hand-side so as to ensure consistency) and the possible upwinding is performed with respect to the material velocity only. The implicit-in-time first-order upwind scheme satisfies a local entropy inequality. A generalization of the convection term is then introduced, which allows to limit the scheme diffusion while ensuring a weaker property: the entropy inequality is satisfied up to a remainder term which is shown to tend to zero with the space and time steps, if the discrete solution is controlled in L $\infty$ and BV norms. The explicit upwind variant also satisfies such a weaker property, at the price of an estimate for the velocity which could be derived from the introduction of a new stabilization term in the momentum balance. Still for the explicit scheme, with the above-mentioned generalization of the convection operator, the same result only holds if the ratio of the time to the space step tends to zero

Compactness of discrete approximate solutions to parabolic PDEs - Application to a turbulence model

International audienceIn this paper, we prove an adaptation of the classical compactness Aubin-Simon lemma to sequences of functions obtained through a sequence of discretizations of a parabolic problem. The main difficulty tackled here is to generalize the classical proof to handle the dependency of the norms controlling each function u (n) of the sequence with respect to n. This compactness result is then used to prove the convergence of a numerical scheme combining finite volumes and finite elements for the solution of a reduced turbulence problem. 1. Introduction. Let us suppose given a sequence of approximations of a parabolic problem (P), which, for instance, may be though of as discretizations of (P) by a numerical scheme, or resulting from the combination of a Faedo-Galerkin technique with a time discretization. In both cases, we are in presence of a family of finite-dimensional systems, and their solution, let us say (u (n)) n∈N , may be considered as a sequence of functions of time, taking their values in a finite dimensional subspace, let us say B (n) of a Banach space (usually a Lebesgue L q space, q ≥ 1). To show the convergence of such a process, a common path is to follow the following steps: 1. first, for each approximate problem, prove the existence of a solution, and derive estimates satisfied by any solution, 2. then use compactness arguments to show (possibly up to the extraction of a subsequence) the existence of a limit, 3. and, finally, prove that this limit satisfies the initial problem (P). Let us now focus on item 2, which is the issue addressed in this paper. The problem here is to prove a compactness result for a sequence of functions of time taking their value in a sequence of discrete spaces and controlled (themselves and their discrete time-derivative) in discrete norms; in the general case, both B (n) and the space part of the norms depend on n. The compactness result given in this paper relies on this particular structure, and consists in a generalization of the classical Aubin-Simon lemma to this specific case

Improved version of the eikonal model for absorbing spherical particles

We present a new expression of the scattering amplitude, valid for spherical absorbing objects, which leads to an improved version of the eikonal method outside the diffraction region. Limitations of this method are discussed and numerical results are presented and compared successfully with the Mie theory.Comment: 7 pages, postscript figures available on cpt.univ-mrs.fr, to appear in J. Mod. Optic

Détermination des substrats lacustres par hydroacoustique : application au suivit de qualité morphologique

La diversité des écosystèmes lacustres s’explique en partie par la variété hydromorphologique des lacs ; la nature et la répartition du substrat qui tapisse leur fond sont des composantes de ce paramètre. C’est pour cela que la Directive Cadre sur l’Eau impose une description de la nature des sédiments des plans d’eau. Plus généralement, la répartition des substrats peut être considérée comme un facteur de structuration des espèces biologiques qu’abrite un lac. Des outils traditionnels comme l’utilisation d’une benne à sédiment ou d’une caméra subaquatique permettent de déterminer la nature des substrats de manière très ponctuelle mais ces techniques atteignent leurs limites lorsque tout un plan d’eau doit être caractérisé. Depuis les années 1980 des méthodes de caractérisation des sédiments utilisant des outils acoustiques qui permettent d’obtenir des informations en continu le long de parcours réalisés par un navire ont été développées et commercialisées. Pour l’application aux écosystèmes lacustres, de part leur mise en oeuvre, les systèmes utilisant les échosondeurs mono‐faisceau paraissent les plus appropriés. Ces appareils permettent de réaliser des cartes de la répartition des substrats à l’échelle du lac. Les informations pertinentes qui permettent de décrire un état biologique potentiel ou de définir un état initial peuvent être ainsi intégrer dans la mise en place des réseaux de suivi de la qualité des milieux. / The lake ecosystem diversity is explained, in part, by the hydromorphological diversity of lakes; nature and repartition of the substrata give information about this parameter. That is the reason why the European Water Framework imposes a substrata nature description of lakes. The substrata repartition could be considered as a factor of the biological structuring presents in lakes. Traditional tools like grab samplers or video cameras enable to determine the substrata nature but they are not appropriate for getting a high resolution description for an entire lake. From the beginning of the 80’s, acoustic devices specialized in seabed classification have been developed and commercialized. For lakebed surveys, systems using single beam sounders seem to be more appropriate; they enable to obtain maps of the lakebed at the whole lake scale. Information which describes a potential biological state can be used for the establishment of a quality monitoring

Thermodynamics of a Heavy Ion-Irradiated Superconductor: the Zero-Field Transition

Specific heat measurements show that the introduction of amorphous columnar defects considerably affects the transition from the normal to the superconducting state in zero magnetic field. Experimental results are compared to numerical simulations of the 3D XY model for both the pure system and the system containing random columnar disorder. The numerics reproduce the salient features of experiment, showing in particular that the specific heat peak changes from cusp-like to smoothly rounded when columnar defects are added. By considering the specific heat critical exponent alpha, we argue that such behavior is consistent with recent numerical work [Vestergren et al., PRB 70, 054508 (2004)] showing that the introduction of columnar defects changes the universality class of the transition.Comment: 4 pages, 2 figure

Yang-Mills gauge anomalies in the presence of gravity with torsion

The BRST transformations for the Yang-Mills gauge fields in the presence of gravity with torsion are discussed by using the so-called Maurer-Cartan horizontality conditions. With the help of an operator \d which allows to decompose the exterior spacetime derivative as a BRST commutator we solve the Wess-Zumino consistency condition corresponding to invariant Chern-Simons terms and gauge anomalies.Comment: 24 pages, report REF. TUW 94-1

Stranding of a humpback whale (Megaptera novaeangliae) on the Belgian coast

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