We introduce a free boundary model to study the effect of vesicle transport onto neurite growth. It consists of systems of drift-diffusion equations describing the evolution of the density of antero- and retrograde vesicles in each neurite coupled to reservoirs located at the soma and the growth cones of the neurites, respectively. The model allows for a change of neurite length as a function of the vesicle concentration in the growth cones. After establishing existence and uniqueness for the time-dependent problem, we briefly comment on possible types of stationary solutions. Finally, we provide numerical studies on biologically relevant scales using a finite volume scheme. We illustrate the capability of the model to reproduce cycles of extension and retraction.

The Wasserstein gradient flow structure of the partial differential equation system governing multiphase flows in porous media was recently highlighted in Cancès et al. [Anal. PDE10(8), 1845–1876]. The model can thus be approximated by means of the minimising movement (or JKO after Jordan, Kinderlehrer and Otto [SIAM J. Math. Anal.29(1), 1–17]) scheme that we solve thanks to the ALG2-JKO scheme proposed in Benamou et al. [ESAIM Proc. Surv.57, 1–17]. The numerical results are compared to a classical upstream mobility finite volume scheme, for which strong stability properties can be established.

In this paper, we propose implicit and semi-implicit in time finite volume schemes forthe barotropic Euler equations (hence, as a particular case, for the shallow waterequations) and for the full Euler equations, based on staggered discretizations. Forstructured meshes, we use the MAC finite volume scheme, and, for general mixedquadrangular/hexahedral and simplicial meshes, we use the discrete unknowns of theRannacher−Turek orCrouzeix−Raviart finiteelements. We first show that a solution to each of these schemes satisfies a discretekinetic energy equation. In the barotropic case, a solution also satisfies a discreteelastic potential balance; integrating these equations over the domain readily yieldsdiscrete counterparts of the stability estimates which are known for the continuousproblem. In the case of the full Euler equations, the scheme relies on the discretizationof the internal energy balance equation, which offers two main advantages: first, we avoidthe space discretization of the total energy, which involves cell-centered andface-centered variables; second, we obtain an algorithm which boils down to a usualpressure correction scheme in the incompressible limit. Consistency (in a weak sense) withthe original total energy conservative equation is obtained thanks to corrective terms inthe internal energy balance, designed to compensate numerical dissipation terms appearingin the discrete kinetic energy inequality. It is then shown in the 1D case, that,supposing the convergence of a sequence of solutions, the limit is an entropy weaksolution of the continuous problem in the barotropic case, and a weak solution in the fullEuler case. Finally, we present numerical results which confirm this theory.

This paper is a numerical work to investigate thermal interactions between heat fluxes generated by electronic devices mounted on a vertical printed circuit board (PCB). In fact, the effect of some parameters such as Grashof number (Gr), the distance (S) between the heat sources and the upper exit distance (Le) was studied. The results show that a regular and uniform heat sources distribution in the inlet is very important in order to take advantage from the leading edge of the boundary layer and to obtain the necessary heat dissipation. The impact of parameters on the heat dissipation, characterized by the Nusselt number, has a different importance. For a Prandtl number Pr = 0.71, the (Gr) increases the heat exchange that is reflected by the increase of Nusselt number, and also participates to the formation of recirculation zones. The total Nusselt number Nu is increased when (Gr) is multiplied by 102. With regards to the distance (S) between the heat sources, the results show that Nu increases about 7% if the distance (S) doubles, and becomes approximately 17.8% when (S) quadruples. At the end, the heat transfer increases when increasing the distance (Le) of the upper exit length, especially on the second component. The total Nusselt number increases by 1.6% when (Le) increases by about 67%.

A new set of boundary conditions has been derived by rigorous methods for the shallow water equations in a limited domain. The aim of this article is to present these boundary conditions and to report on numerical simulations which have been performed using these boundary conditions. The new boundary conditions which are mildly dissipative let the waves move freely inside and outside the domain. The problems considered include a one-dimensional shallow water system with two layers of fluids and a two-dimensional inviscid shallow water system in a rectangle.

After we derive the Serre system of equations of water wave theory from a generalized variational principle, we present some of its structural properties. We also propose a robust and accurate finite volume scheme to solve these equations in one horizontal dimension. The numerical discretization is validated by comparisons with analytical and experimental data or other numerical solutions obtained by a highly accurate pseudo-spectral method.

In this work we introduce a new class of lowest order methods for diffusive problems on general meshes with only one unknown per element. The underlying idea is to construct an incomplete piecewise affine polynomial space with optimal approximation properties starting from values at cell centers. To do so we borrow ideas from multi-point finite volume methods, although we use them in a rather different context. The incomplete polynomial space replaces classical complete polynomial spaces in discrete formulations inspired by discontinuous Galerkin methods. Two problems are studied in this work: a heterogeneous anisotropic diffusion problem, which is used to lay the pillars of the method, and the incompressible Navier-Stokes equations, which provide a more realistic application. An exhaustive theoretical study as well as a set of numerical examples featuring different difficulties are provided.

We present in this paper a pressure correction scheme for the drift-flux model combining finite element and finite volume discretizations, which is shown to enjoy essential stability features of the continuous problem: the scheme is conservative, the unknowns are kept within their physical bounds and, in the homogeneous case (i.e. when the drift velocity vanishes), the discrete entropy of the system decreases; in addition, when using for the drift velocity a closure law which takes the form of a Darcy-like relation, the drift term becomes dissipative.Finally, the present algorithm preserves a constant pressure and a constant velocity through moving interfaces between phases.To ensure the stability as well as to obtain this latter property, a key ingredient is to couple the mass balance and the transport equation for the dispersed phase in an original pressure correction step.The existence of a solution to each step of the algorithm is proven; in particular, the existence of a solution to the pressure correction step is derived as a consequence of a more general existence result for discrete problems associated to the drift-flux model.Numerical tests show a near-first-order convergence rate for the scheme, both in time and space, and confirm its stability.

We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows.Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes.Under a simple geometrical assumption for the clusters, we obtain that the pair of discrete spaces associating the classical cell-centered approximation for the velocities and cluster-wide constant pressures is inf-sup stable; in addition, we prove that a stabilization involving pressure jumps only across the internal edges of the clusters yields a stable scheme with the usual collocated discretization (i.e. , in particular, with control-volume-wide constant pressures), for the Stokes and the Navier-Stokes problem.An analysis of this stabilized scheme yields the existence of the discrete solution (and uniqueness for the Stokes problem). The convergence of the approximate solution toward the solution to the continuous problem as the mesh size tends to zero is proven, provided, in particular, that the approximation of the mass balance flux is second order accurate; this condition imposes some geometrical conditions on the mesh.Under the same assumption, an error analysis is provided for the Stokes problem: it yields first-order estimates in energy norms.Numerical experiments confirm the theory and show, in addition, a second order convergence for the velocity in a discrete L2 norm.

We present and analyse in this paper a novel colocated Finite Volume scheme for the solution of the Stokes problem.It has been developed following two main ideas.On one hand, the discretization of the pressure gradient term is built as the discrete transposed of the velocity divergence term, the latter being evaluated using a natural finite volume approximation; this leads to a non-standard interpolation formula for the expression of the pressure on the edges of the control volumes.On the other hand, the scheme is stabilized using a finite volume analogue to the Brezzi-Pitkäranta technique.We prove that, under usual regularity assumptions for the solution (each component of the velocity in H2(∞) and pressure in H1(∞)), the scheme is first order convergent in the usual finite volume discrete H1 norm and the L2 norm for respectively the velocity and the pressure, provided, in particular, that the approximation of the mass balance flux is of second order.With the above-mentioned interpolation formulae, this latter condition is satisfied only for particular meshes: acute angles triangulations or rectangular structured discretizations in two dimensions, and rectangular parallelepipedic structured discretizations in three dimensions.Numerical experiments confirm this analysis and show, in addition, a second order convergence for the velocity in a discrete L2 norm.

This paper is devoted to the numerical simulation of wavebreaking. It presents the results of a numerical workshop that washeld during the conference LOMA04. The objective is to compareseveral mathematical models (compressible or incompressible) andassociated numerical methods to compute the flow field during awave breaking over a reef. The methods will also be compared withexperiments.

We analyze residual and hierarchicala posteriori error estimates for nonconforming finite elementapproximations of elliptic problems with variable coefficients.We consider a finite volume box scheme equivalent toa nonconforming mixed finite element method in a Petrov–Galerkinsetting. We prove thatall the estimators yield global upper and local lower bounds for the discretizationerror. Finally, we present results illustrating the efficiency of theestimators, for instance, in the simulation of Darcy flows throughheterogeneous porous media.

Ce travail concerne l'étude des instabilités convectives et l'interaction solide/liquide pour une configuration de croissance dirigée horizontale d'un matériau à faible nombre de Prandtl (Pr=0,015). Les configurations de ce type ont fait l'objet de plusieurs études restreintes à la phase fluide. Il s'agit ici d'étudier les transitions au sein du bain fondu en présence de la convection thermocapillaire et l'interaction avec le front. L'interaction se focalisant dans cette étude sur les aspects convectifs sans prise en compte des effets de la chaleur latente. L'approche de résolution est basée sur la méthode de localisation du front couplée à une approximation de type volumes finis. La convection thermocapillaire agissant dans le même sens que la convection thermogravitationnelle amorti les oscillations introduites par la convection thermogravitationnelle. D'autre part, l'effet de la vitesse de translation de l'ampoule pour le contrôle du processus est également analysé.

We prove the convergence of a finite volumemethod for a noncoercive linear elliptic problem, with right-handside in the dual space of the natural energy space of the problem.

We study a finite volume method, used to approximate the solution of the linear two dimensional convection diffusion equation, with mixed Dirichlet and Neumann boundary conditions, on Cartesian meshes refined by an automatic technique (which leads to meshes with hanging nodes). We propose an analysis through a discrete variational approach, in a discrete H 1 finite volume space. We actually prove the convergence of the scheme in a discrete H 1 norm, with an error estimate of order O(h) (on meshes of size h).

In this paper, a class of cell centered finite volume schemes, on general unstructured meshes, for a linear convection-diffusion problem, is studied. The convection and the diffusion are respectively approximated by means of an upwind scheme and the so called diamond cell method [4]. Our main result is an error estimate oforder h, assuming only the W2,p (for p>2) regularity of thecontinuous solution, on a mesh of quadrangles. The proof is based on anextension of the ideas developed in [12]. Some newdifficulties arise here, due to the weak regularity of the solution, and thenecessity to approximate the entire gradient, and not only its normalcomponent, as in [12].