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Nash equilibria for a model of traffic flow with several groups of drivers

Alberto Bressan, Ke Han (2012)

ESAIM: Control, Optimisation and Calculus of Variations

Traffic flow is modeled by a conservation law describing the density of cars. It is assumed that each driver chooses his own departure time in order to minimize the sum of a departure and an arrival cost. There are N groups of drivers, The i-th group consists of κi drivers, sharing the same departure and arrival costs ϕi(t),ψi(t). For any given population sizes κ1,...,κn, we prove the existence of a Nash equilibrium solution, where no driver can lower his own total cost by choosing a different departure...

New semi-Hamiltonian hierarchy related to integrable magnetic flows on surfaces

Misha Bialy, Andrey Mironov (2012)

Open Mathematics

We consider magnetic geodesic flows on the two-torus. We prove that the question of existence of polynomial in momenta first integrals on one energy level leads to a semi-Hamiltonian system of quasi-linear equations, i.e. in the hyperbolic regions the system has Riemann invariants and can be written in conservation laws form.

Non-uniqueness of almost unidirectional inviscid compressible flow

Pavel Šolín, Karel Segeth (2004)

Applications of Mathematics

Our aim is to find roots of the non-unique behavior of gases which can be observed in certain axisymmetric nozzle geometries under special flow regimes. For this purpose, we use several versions of the compressible Euler equations. We show that the main reason for the non-uniqueness is hidden in the energy decomposition into its internal and kinetic parts, and their complementary behavior. It turns out that, at least for inviscid compressible flows, a bifurcation can occur only at flow regimes with...

Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime

Rémi Carles, Bijan Mohammadi (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We study numerically the semiclassical limit for the nonlinear Schrödinger equation thanks to a modification of the Madelung transform due to Grenier. This approach allows for the presence of vacuum. Even if the mesh size and the time step do not depend on the Planck constant, we recover the position and current densities in the semiclassical limit, with a numerical rate of convergence in accordance with the theoretical results, before shocks appear in the limiting Euler equation. By using simple...

Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime

Rémi Carles, Bijan Mohammadi (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We study numerically the semiclassical limit for the nonlinear Schrödinger equation thanks to a modification of the Madelung transform due to Grenier. This approach allows for the presence of vacuum. Even if the mesh size and the time step do not depend on the Planck constant, we recover the position and current densities in the semiclassical limit, with a numerical rate of convergence in accordance with the theoretical results, before shocks appear in the limiting Euler equation. By using simple...

Numerical boundary layers for hyperbolic systems in 1-D

Claire Chainais-Hillairet, Emmanuel Grenier (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The aim of this paper is to investigate the stability of boundary layers which appear in numerical solutions of hyperbolic systems of conservation laws in one space dimension on regular meshes. We prove stability under a size condition for Lax Friedrichs type schemes and inconditionnal stability in the scalar case. Examples of unstable boundary layers are also given.

Numerical boundary layers for hyperbolic systems in 1-D

Claire Chainais-Hillairet, Emmanuel Grenier (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The aim of this paper is to investigate the stability of boundary layers which appear in numerical solutions of hyperbolic systems of conservation laws in one space dimension on regular meshes. We prove stability under a size condition for Lax Friedrichs type schemes and inconditionnal stability in the scalar case. Examples of unstable boundary layers are also given.

Numerical flux-splitting for a class of hyperbolic systems with unilateral constraint

Florent Berthelin (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We study in this paper some numerical schemes for hyperbolic systems with unilateral constraint. In particular, we deal with the scalar case, the isentropic gas dynamics system and the full-gas dynamics system. We prove the convergence of the scheme to an entropy solution of the isentropic gas dynamics with unilateral constraint on the density and mass loss. We also study the non-trivial steady states of the system.

Numerical flux-splitting for a class of hyperbolic systems with unilateral constraint

Florent Berthelin (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We study in this paper some numerical schemes for hyperbolic systems with unilateral constraint. In particular, we deal with the scalar case, the isentropic gas dynamics system and the full-gas dynamics system. We prove the convergence of the scheme to an entropy solution of the isentropic gas dynamics with unilateral constraint on the density and mass loss. We also study the non-trivial steady states of the system.

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