We consider a class of nonlocal crowd dynamics models for N populations with different destinations trying to avoid each other in a confined walking domain.
This can be formalized in a initial-boundary value problem for a system of nonlocal conservation laws, where the velocity vector field of each population depends on a nonlocal operator depending on the current density distribution.
To...
This talk concerns the numerical approximations of the weak solutions of scalar hyperbolic conservation laws. After showing how to bypass the barrier theorems for the linear advection, the derivation of a second-order entropy-stable scheme will be presented for non-linear equations. The fully discrete stability result will be established for regular strictly convex entropy and under a...
We introduce a relaxation system to approximate the solutions to the barotropic Euler equations. We show that the solutions to this two-speed relaxation model can be understood as viscous approximations of the solutions to the barotropic Euler equations under appropriate sub-characteristic conditions. Our relaxation system is a generalization of the well-known Suliciu relaxation system, and it...
The method of moments is commonly used to reduce a kinetic equation into a fluid model. It can be seen as a semi-discretization with respect to the kinetic variable, and it results in a system of balance laws that needs additional closure relations. The choices made in this construction have impacts on the properties of the resulting system, namely the strong or weak hyperbolicity, the entropy...
We are interested in 2 × 2 systems of conservation laws of special structure, including
generalized Aw-Rascle and Zhang (GARZ) models for road traffic. The simplest representative
is the Keytz-Kranzer system, where one equation is nonlinear and not coupled to the other, and
the second equation is a linear transport which coefficients depend on the solution of the firstequation.
In GARZ...
When considering the numerical approximation of weak solutions of systems of conservation laws with source term, the satisfaction of discrete entropy inequalities is, in general, very difficult to be obtained. In the present talk, we present a suitable control of the artificial numerical viscosity in order to recover the expected discrete entropy inequalities. Moreover, the artificial...
Le modèle pour évacuation piétonnière proposé par R.L. Hughes au début des années 2000 combine l'évacuation des agents en suivant la dynamique d'une loi de conservation avec champ de vitesses discontinu et l'ajustement instantané dudit champ pénalisant les régions à haute densité d'agents. L'analyse mathématique de cet élégant modèle reste délicate, à cause des singularités de chacune des...
Our goal is to introduce a mathematical model for gas flow through a one-way valve which opens and closes (any value between fully open and fully closed is attainable) depending on the gap between the upstream and the downstream values of the pressure. This can be done by representing the valve as a unilateral point constraint on the gas flow, and introducing a suitable non-classical Riemann...
We are concerned with the existence and compactness of entropy solutions of the compressible Euler system for two dimensional steady potential flow around an obstacle for a polytropic gas with supersonic far-field velocity. This problem was first formulated by Morawetz in 1985 and has remained open since then. In this paper, we develop a complete compactness framework that allows for...
In this talk we first discuss the different plasma-vacuum interface problems of ideal Magneto-Hydrodynamics for incompressible or compressible fluids. Then we focus on the plasma-vacuum interface problem for ideal relativistic Magneto-Hydrodynamics (RMHD) in three-dimensional Minkowski spacetime. The plasma flow is governed by the two-dimensional RMHD equations, while the vacuum magnetic and...
This talk deals with the stability analysis of discrete shock profiles
for systems of conservation laws. These profiles correspond to
approximations of shocks of systems of conservation laws by
conservative finite difference schemes. Discontinuous solutions
appear naturally in the study of systems of conservation laws,
which can model many physical situations, such as gas...
This study focuses on the development of numerical schemes for a monolithic hyperbolic Eulerian model that combines fluid and hyper-elastic solid dynamics. Solid behavior requires an additional equation for deformation, and traditional approaches represent this model variable using the gradient of the backward characteristics rather than the characteristics themselves. While this ensures a...
The approximate resolution of hyperbolic problems using finite difference methods requires a specific treatment of boundary conditions: artificial truncation of the computational domain, incorporation of physical boundary conditions, and so on. Even if the interior scheme has good convergence properties, the choice of boundary scheme can seriously impair the quality of the overall...
We will discuss the shock profiles for the Lax–Wendroff scheme. This numerical scheme constructs solutions to systems of conservation laws, with shock waves providing the simplest example of discontinuous solutions. In the case of scalar conservation laws, we will recall a result on the existence of shock profiles for this scheme and study their stability. This is a joint result with...
