Numerical scheme for multilayer shallow-water model in the low-Froude number regime

International audienceThe aim of this note is to present a multi-dimensional numerical scheme approximating the solutions of the multilayer shallow water model in the low Froude number regime. The proposed strategy is based on a regularized model where the advection velocity is modified with a pressure gradient in both mass and momentum equations. The numerical solution satisfy the dissipation of energy, which act for mathematical entropy, and the main physical properties required for simulations within oceanic flows. Résumé Schéma numérique pour lesmo eles de Saint-Venant multi-couchè a faible nombre de Froude. Le but de cette note est de présenter un schéma numérique multi-dimensionnel rapprochant les solutions dumo ele de Saint-Venant multi-couche en régime de faible nombre de Froude. La stratégie proposée est basée sur unmo ele régulariséò u la vitesse de transport est modifié par un gradient de pression dans le equations de la masse et de la quantité de mouvement. La solution numérique satisfait la dissipation denergie,jouantlerôledel'entropiedupointdevuemathématique,etlesprincipalespropriétésphysiquesnécessairesauxsimulationsdanslecadredeecoulementsocéanique

Centered-potential regularization for the advection upstream splitting method

International audienceThis paper is devoted to a centered IMEX scheme in a multidimensional framework for a wide class of multicomponent and isentropic flows. The proposed strategy is based on a regularized model where the advection velocity is modified by the gradient of the potential of the conservative forces in both mass and momentum equations. The stability of the scheme is ensured by the dissipation of mechanic energy, which stands for a mathematical entropy, under an advective CFL condition. The main physical properties, such as positivity, conservation of the total momentum, and conservation of the steady state at rest, are satisfied. In addition, asymptotic preserving properties in the regimes (“incompressible” and “acoustic”) are analyzed. Finally, several simulations are presented to illustrate our results in a simplified context of oceanic flows in one dimension

Developmental Regulation and Spatiotemporal Redistribution of the Sumoylation Machinery in the Rat Central Nervous System

International audienceBACKGROUND: Small Ubiquitin-like MOdifier protein (SUMO) is a key regulator of nuclear functions but little is known regarding the role of the post-translational modification sumoylation outside of the nucleus, particularly in the Central Nervous System (CNS). METHODOLOGY/PRINCIPAL FINDINGS: Here, we report that the expression levels of SUMO-modified substrates as well as the components of the sumoylation machinery are temporally and spatially regulated in the developing rat brain. Interestingly, while the overall sumoylation is decreasing during brain development, there are progressively more SUMO substrates localized at synapses. This increase is correlated with a differential redistribution of the sumoylation machinery into dendritic spines during neuronal maturation. CONCLUSIONS/SIGNIFICANCE: Overall, our data clearly demonstrate that the sumoylation process is developmentally regulated in the brain with high levels of nuclear sumoylation early in the development suggesting a role for this post-translational modification during the synaptogenesis period and a redistribution of the SUMO system towards dendritic spines at a later developmental stage to modulate synaptic protein function

A simple kinetic equation of swarm formation: blow–up and global existence

International audienceIn the present paper we identify both blow–up and global existence behaviors for a simple but very rich kinetic equation describing of a swarm formation

Intracellular protein dynamics as a mathematical problem

International audienceIn this paper we undertake a mathematical analysis of a model of intracellular protein dynamics , i.e. protein and mRNA transport inside a cell, proposed by Szymanska at al. in 2014. The model takes into account diffusive transport in the nucleus and cytoplasm, as well as active transport of protein molecules along microtubules in the cytoplasm. The model reproduces, at least in numerical simulations, the oscillatory changes in protein concentration observed in the experimental data. To our knowledge this is the first paper that, in the multidimensional case, deals with a rigorous mathematical analysis of a model of intracellular dynamics with active transport on microtubules. In particular, in the present paper we prove well-posedness of the model in any space dimension. The model is a complex system of nonlinear PDEs with specific boundary conditions. It may be adapted to other signaling pathways

Blow-up and global existence for a kinetic equation of swarm formation

International audienceIn the present paper we study possible blow–ups and global existence for a kinetic equation that describes swarm formations in the variable interacting rate case

Congested shallow water model: on floating body

International audienceWe consider the floating body problem in the vertical plane on a large space scale. More precisely, we are interested in the numerical modeling of body floating freely on the water such as icebergs or wave energy converters.The fluid-solid interaction is formulated using a congested shallow water model for the fluid and Newton's second law of motion for the solid. We make a particular focus on the energy transfer between the solid and the water since it is of major interest for energy production. A numerical approximation based on the coupling of a nite volume scheme for the fluid and a Newmark scheme for the solid is presented. An entropy correction based on an adapted choice of discretization for the coupling terms is made in order to ensure a dissipation law at the discrete level. Simulations are presented to verify the method and to show the feasibility of extending it to more complex cases

Congested shallow water model: roof modelling in free surface flow

International audienceWe are interested in the modeling and the numerical approximation of flows in the presence of a roof, for example flows in sewers or under an ice floe. A shallow water model with a supplementary congestion constraint describing the roof is derived from the Navier-Stokes equations. The congestion constraint is a challenging problem for the numerical resolution of hyperbolic equations. To overcome this difficulty, we follow a pseudo-compressibility relaxation approach. Eventually, a numerical scheme based on a Finite Volume method is proposed. The well-balanced property and the dissipation of the mechanical energy, acting as a mathematical entropy, are ensured under a non-restrictive condition on the time step in spite of the large celerity of the potential waves in the congested areas. Simulations in one dimension for transcritical steady flow are carried out and numerical solutions are compared to several analytical (stationary and non-stationary) solutions for validation

A hierarchy of dispersive layer-averaged approximations of Euler equations for free surface flows

International audienceIn geophysics, the shallow water model is a good approximation of the incompressible Navier-Stokes system with free surface and it is widely used for its mathematical structure and its computational efficiency. However, applications of this model are restricted by two approximations under which it was derived, namely the hydrostatic pressure and the vertical averaging. Each approximation has been addressed separately in the literature: the first one was overcome by taking into account the hydrodynamic pressure (e.g. the non-hydrostatic or the Green-Naghdi models); the second one by proposing a multilayer version of the shallow water model.In the present paper, a hierarchy of new models is derived with a layerwise approach incorporating non-hydrostatic effects to model the Euler equations. To assess these models, we use a rigorous derivation process based on a Galerkin-type approximation along the vertical axis of the velocity field and the pressure, it is also proven that all of them satisfy an energy equality. In addition, we analyse the linear dispersion relation of these models and prove that the latter relations converge to the dispersion relation for the Euler equations when the number of layers goes to infinity