### Singular trajectories and subanalyticity in optimal control and Hamilton-Jacobi theory.

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In this survey paper we present recent advances in some classes of differential game in which there is an asymmetry of information between the players. We explain that—under suitable structure conditions—these games have a value, which can be characterized in terms of (new) Hamilton-Jacobi equations.

We are interested in simulating blood flow in arteries with variable elasticity with a one dimensional model. We present a well-balanced finite volume scheme based on the recent developments in shallow water equations context. We thus get a mass conservative scheme which also preserves equilibria of = 0. This numerical method is tested on analytical tests.

This paper focuses on the contribution of the second order corrector in periodic homogenization applied to a conductive-radiative heat transfer problem. Especially, for a heat conduction problem in a periodically perforated domain with a non-local boundary condition modelling the radiative heat transfer, if this model contains an oscillating thermal source and a thermal exchange with the perforations, the second order corrector helps us to model the gradients which appear between the source area...

For the one-dimensional Schrödinger equation, some real intervals with no eigenvalues (the spectral gaps) may be obtained rather systematically with a method proposed by H. Giacomini and A. Mouchet in 2007. The present article provides some alternative formulation of this method, suggests some possible generalizations and extensively discusses the higher-dimensional case.

This paper describes a numerical method for the two-dimensional sine-Gordon equation over a rectangular domain using differentiation matrices, in the theoretical frame of matrix differential equations.

The asymptotic description of the semiclassical limit of nonlinear Schrödinger equations is a major challenge with so far only scattered results in 1 + 1 dimensions. In this limit, solutions to the NLS equations can have zones of rapid modulated oscillations or blow up. We numerically study in this work the Davey-Stewartson system, a 2 + 1 dimensional nonlinear Schrödinger equation with a nonlocal term, by using parallel computing. This leads to the first results on the semiclassical limit for...

The Poisson-Boltzmann (PB) equation describes the electrostatic potential of a biomolecular system composed by a molecule in a solvent. The electrostatic potential is involved in biomolecular models which are used in molecular simulation. In consequence, finding an efficient method to simulate the numerical solution of PB equation is very useful. As a first step, we establish in this paper a probabilistic interpretation of the nonlinear PB equation with Backward Stochastic Differential Equations...

We study the Maxwell-Landau-Lifshitz system for highly oscillating initial data, with characteristic frequencies (1 ) and amplitude (1), over long time intervals (1 ), in the limit → 0. We show that a nonlinear Schrödinger equation gives a good approximation for the envelope of the solution in the time interval under consideration. This extends previous results of Colin and Lannes [1]. This text is a short version of the article [5].

We propose a method dedicated to the simulation of interface flows involving an arbitrary number of compressible components. Our task is two-fold: we first introduce a -component flow model that generalizes the two-material five-equation model of [2,3]. Then, we present a discretization strategy by means of a Lagrange-Remap [8,10] approach following the lines of [5,7,12]. The projection step involves an anti-dissipative mechanism derived from [11,12]. This feature allows to prevent the numerical...

We propose a finite volume scheme to study the oscillations of a spherical bubble of gas in a liquid phase. Spherical symmetry implies a geometric source term in the Euler equations. Our scheme satisfies the well-balanced property. It is based on the VFRoe approach. In order to avoid spurious pressure oscillations, the well-balanced approach is coupled with an ALE (Arbitrary Lagrangian Eulerian) technique at the interface and a random sampling remap.

We propose a finite volume method on general meshes for the numerical simulation of an incompressible and immiscible two-phase flow in porous media. We consider the case that can be written as a coupled system involving a degenerate parabolic convection-diffusion equation for the saturation together with a uniformly elliptic equation for the global pressure. The numerical scheme, which is implicit in time, allows computations in the case of a heterogeneous...

We are studying an optimal control problem with free initial condition. The initial state of the optimized system is not known exactly, information on initial state is exhausted by inclusions ∈ . Accessible controls for optimization of continuous dynamic system are discrete controls defined on quantized axes. The method presented is based on the concepts and operations of the adaptive method [9] of linear programming. The results are illustrated by a fourth order...

The equations of physics are mathematical models consisting of geometric objects and relationships between then. There are many methods to discretize equations, but few maintain the physical nature of objects that constitute them. To respect the geometrical nature elements of physics, it is necessary to change the point of view and using differential geometry, including the numerical study. We propose to construct discrete differential forms using B-splines and a formulation discrete for different...

Some new iterative techniques are defined to solve reversible inverse problems and a common formulation is explained. Numerical improvements are suggested and tests validate the methods.

We present a basic althought little known numerical stability condition: for convection equations, the von Neumann stability constraint ∥ ∥ ≤ (1 + Δ) ∥ ∥ drives to the stability condition Δ ≤ Δ with $\mathit{\alpha}\mathrm{=}\frac{\mathit{p}\mathrm{(}\mathrm{2}\mathit{q}\mathrm{-}\mathrm{1}\mathrm{)}}{\mathit{q}\mathrm{(}\mathrm{2}\mathit{p}\mathrm{-}\mathrm{1}\mathrm{)}}$ where is an integer linked to the stability domain of the time scheme and ≥ an integer linked to the upwind property of the space discretization (in the...

We propose to formally derive a low Mach number model adapted to the modeling of a water nuclear core ( of PWR- or BWR-type) in the forced convection regime or in the natural convection regime by filtering out the acoustic waves in the compressible Navier-Stokes system. Then, we propose a monodimensional stationary analytical solution with regular and singular charge loss when the equation of state is a stiffened gas equation. Moreover, we show that...

High time frequency oscillations occur in many different physical cases: slightly compressible fluids, almost quasineutral plasmas, small electron mass approximation .... In many case, small parameters arise in fluids mechanics or plasma physics, leading to these oscillations as the small parameter goes to zero. The aim of this note is to detail how to obtain formal expansions and to give some indications on how to justify them.

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