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AbstractAbstract

[en] In this paper, we are interested in modelling the flow of the coolant (water) in a nuclear reactor core. To this end, we use a mono dimensional low Mach number model supplemented with the stiffened gas law. We take into account potential phase transitions by a single equation of state which describes both pure and mixture phases. In some particular cases, we give analytical steady and/or unsteady solutions which provide qualitative information about the flow. In the second part of the paper, we introduce two variants of a numerical scheme based on the method of characteristics to simulate this model. We study and verify numerically the properties of these schemes. We finally present numerical simulations of a loss of flow accident (LOFA) induced by a coolant pump trip event. (authors)

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Available from doi: http://dx.doi.org/10.1051/m2an/2014015; 44 refs.

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Mathematical Modelling and Numerical Analysis; ISSN 0764-583X; ; v. 48; p. 1639-1679

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[en] We consider an initial-boundary value problem for a generalized 2D time-dependent Schroedinger equation (with variable coefficients) on a semi-infinite strip. For the Crank-Nicolson-type finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the unconditional uniform in time L2-stability is proved. Due to the splitting, an effective direct algorithm using FFT is developed now to implement the method with the discrete TBC for general potential. Numerical results on the tunnel effect for rectangular barriers are included together with the detailed practical error analysis confirming nice properties of the method. (authors)

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Available from doi: http://dx.doi.org/10.1051/m2an/2014004; 36 refs.

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Mathematical Modelling and Numerical Analysis; ISSN 0764-583X; ; v. 48; p. 1681-1699

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[en] We derive an a posteriori error estimation for the discrete duality finite volume (DDFV) discretization of the stationary Stokes equations on very general two-dimensional meshes, when a penalty term is added in the incompressibility equation to stabilize the variational formulation. Two different estimators are provided: one for the error on the velocity and one for the error on the pressure. They both include a contribution related to the error due to the stabilization of the scheme, and a contribution due to the discretization itself. The estimators are globally upper as well as locally lower bounds for the errors of the DDFV discretization. They are fully computable as soon as a lower bound for the inf-sup constant is available. Numerical experiments illustrate the theoretical results and we especially consider the influence of the penalty parameter on the error for a fixed mesh and also of the mesh size for a fixed value of the penalty parameter. A global error reducing strategy that mixes the decrease of the penalty parameter and adaptive mesh refinement is described. (authors)

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Available from doi: < 32 refs.

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Mathematical Modelling and Numerical Analysis; ISSN 0764-583X; ; v. 49; p. 663-693

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[en] In the present work we investigate the numerical simulation of liquid-vapor phase change in compressible flows. Each phase is modeled as a compressible fluid equipped with its own equation of state (EOS). We suppose that inter-phase equilibrium processes in the medium operate at a short time-scale compared to the other physical phenomena such as convection or thermal diffusion. This assumption provides an implicit definition of an equilibrium EOS for the two-phase medium. Within this framework, mass transfer is the result of local and instantaneous equilibria between both phases. The overall model is strictly hyperbolic. We examine properties of the equilibrium EOS and we propose a discretization strategy based on a finite-volume relaxation method. This method allows to cope with the implicit definition of the equilibrium EOS, even when the model involves complex EOS's for the pure phases. We present two-dimensional numerical simulations that shows that the model is able to reproduce mechanism such as phase disappearance and nucleation. (authors)

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Available from doi: http://dx.doi.org/10.1051/m2an/2011069; 55 refs.

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Mathematical Modelling and Numerical Analysis; ISSN 0764-583X; ; v. 46(no.5); p. 1029-1054

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[en] Cell-centered and vertex-centered finite volume schemes for the Laplace equation with homogeneous Dirichlet boundary conditions are considered on a triangular mesh and on the Voronoi diagram associated to its vertices. A broken P

^{1}function is constructed from the solutions of both schemes. When the domain is two-dimensional polygonal convex, it is shown that this reconstruction converges with second-order accuracy towards the exact solution in the L^{2}norm, under the sufficient condition that the right-hand side of the Laplace equation belongs to H^{1}(Ω). (authors)Primary Subject

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Available from doi: http://dx.doi.org/10.1051/m2an/2010068; 37 refs.

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Mathematical Modelling and Numerical Analysis; ISSN 0764-583X; ; v. 45(no.4); p. 627-650

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[en] This paper develops some new variational principles for the solutions of Hartree Eigen problems and uses these characterizations to describe convergent iterative algorithms for these problems. This is done first for-helium and then for general atoms and molecules. The variational principles involve minimizing separately convex functionals over the product of convex sets. By minimizing in different variables at each step, we are led to descent methods where at each step there is a strictly convex problem with a unique solution. The resulting sequence is shown to converge to a solution of the Hartree Eigen problem. (authors). 23 refs

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[en] The feasibility of the Finite Volume method using a Van Leer scheme on irregular meshes made of quadrilaterals or triangles is shown for the Euler 2D equations. The results are compared with those of a first order scheme. (Author). 11 refs., 12 figs

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[en] We consider a two-dimensional Magnetohydrodynamic system of equations describing the motion of a conducting fluid in which eddy currents flow. The mathematical model is derived and existence of solutions is proved by using fixed point techniques. Uniqueness is obtained under restrictive conditions on the involved physical parameters

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[en] We analyse a quantum-mechanical model for the transport of electrons in semiconductors. The model consists of the quantum Liouville (Wigner) equation posed on the bounded Brillouin zone corresponding to the semiconductor crystal lattice, with a self-consistent potential determined by a Poisson equation. A global existence and uniqueness proof for this model is the main result of the paper

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[en] One gives a new version of the variational principle δL = 0, L being the usual Lagrangian, for the perfect fluid mechanics. It is formally equivalent to the well-known principle but it gives the first rigorous derivation of the conservation laws (momentum and energy), including the discontinuous case (shock waves, contact discontinuities). Thanks to a new formulation of the constraints, we do not involve any Lagrange multiplier, which in previous works were neither physically relevant, since they do not appear in the Euler equations, nor mathematically relevant. We even give a variational interpretation of the entropy inequality when shock waves occur. Our method covers all aspects of the perfect fluids, including stationary and unstationary motion, compressible and incompressible fluids, axisymmetric case. When the velocity field admits a stream function, the variational principle gives rise to extremal points of the Lagrangian on various infinite dimensional manifolds. For a suitable choice of this manifold, the flow is itself periodic, that is all the fluid particles have a periodic motion with the same period. The flow describes a closed geodesic on some group of diffeomorphisms. (author). 10 refs

Original Title

Sur le principe variationnel des equations de la mecanique des fluides parfaits

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