1,513 research outputs found
Numerical simulations of the Euler system with congestion constraint
In this paper, we study the numerical simulations for Euler system with
maximal density constraint. This model is developed in [1, 3] with the
constraint introduced into the system by a singular pressure law, which causes
the transition of different asymptotic dynamics between different regions. To
overcome these difficulties, we adapt and implement two asymptotic preserving
(AP) schemes originally designed for low Mach number limit [2,4] to our model.
These schemes work for the different dynamics and capture the transitions well.
Several numerical tests both in one dimensional and two dimensional cases are
carried out for our schemes
Finite Volume approximations of the Euler system with variable congestion
We are interested in the numerical simulations of the Euler system with
variable congestion encoded by a singular pressure. This model describes for
instance the macroscopic motion of a crowd with individual congestion
preferences. We propose an asymptotic preserving (AP) scheme based on a
conservative formulation of the system in terms of density, momentum and
density fraction. A second order accuracy version of the scheme is also
presented. We validate the scheme on one-dimensional test-cases and extended
here to higher order accuracy. We finally carry out two dimensional numerical
simulations and show that the model exhibit typical crowd dynamics
Two-way multi-lane traffic model for pedestrians in corridors
We extend the Aw-Rascle macroscopic model of car traffic into a two-way
multi-lane model of pedestrian traffic. Within this model, we propose a
technique for the handling of the congestion constraint, i.e. the fact that the
pedestrian density cannot exceed a maximal density corresponding to contact
between pedestrians. In a first step, we propose a singularly perturbed
pressure relation which models the fact that the pedestrian velocity is
considerably reduced, if not blocked, at congestion. In a second step, we carry
over the singular limit into the model and show that abrupt transitions between
compressible flow (in the uncongested regions) to incompressible flow (in
congested regions) occur. We also investigate the hyperbolicity of the two-way
models and show that they can lose their hyperbolicity in some cases. We study
a diffusive correction of these models and discuss the characteristic time and
length scales of the instability
A congestion model for cell migration
This paper deals with a class of macroscopic models for cell migration in a
saturated medium for two-species mixtures. Those species tend to achieve some
motion according to a desired velocity, and congestion forces them to adapt
their velocity. This adaptation is modelled by a correction velocity which is
chosen minimal in a least-square sense. We are especially interested in two
situations: a single active species moves in a passive matrix (cell migration)
with a given desired velocity, and a closed-loop Keller-Segel type model, where
the desired velocity is the gradient of a self-emitted chemoattractant. We
propose a theoretical framework for the open-loop model (desired velocities are
defined as gradients of given functions) based on a formulation in the form of
a gradient flow in the Wasserstein space. We propose a numerical strategy to
discretize the model, and illustrate its behaviour in the case of a prescribed
velocity, and for the saturated Keller-Segel model
Soft congestion approximation to the one-dimensional constrained Euler equations
This article is concerned with the analysis of the one-dimensional
compressible Euler equations with a singular pressure law, the so-called hard
sphere equation of state. The result is twofold. First, we establish the
existence of bounded weak solutions by means of a viscous regularization and
refined compensated compactness arguments. Second, we investigate the smooth
setting by providing a detailed description of the impact of the singular
pressure on the breakdown of the solutions. In this smooth framework, we
rigorously justify the singular limit towards the free-congested Euler
equations, where the compressible (free) dynamics is coupled with the
incompressible one in the constrained (i.e. congested) domain
Self-Organized Hydrodynamics with congestion and path formation in crowds
A continuum model for self-organized dynamics is numerically investigated.
The model describes systems of particles subject to alignment interaction and
short-range repulsion. It consists of a non-conservative hyperbolic system for
the density and velocity orientation. Short-range repulsion is included through
a singular pressure which becomes infinite at the jamming density. The singular
limit of infinite pressure stiffness leads to phase transitions from
compressible to incompressible dynamics. The paper proposes an
Asymptotic-Preserving scheme which takes care of the singular pressure while
preventing the breakdown of the CFL stability condition near congestion. It
relies on a relaxation approximation of the system and an elliptic formulation
of the pressure equation. Numerical simulations of impinging clusters show the
efficiency of the scheme to treat congestions. A two-fluid variant of the model
provides a model of path formation in crowds
One-dimensional granular system with memory effects
We consider a hybrid compressible/incompressible system with memory effects
introduced by Lefebvre Lepot and Maury (2011) for the description of
one-dimensional granular flows. We prove a first global existence result for
this system without additional viscous dissipation. Our approach extends the
one by Cavalletti, Sedjro, Westdickenberg (2015) for the pressureless Euler
system to the constraint granular case with memory effects. We construct
Lagrangian solutions based on an explicit formula of the monotone rearrangement
associated to the density and explain how the memory effects are linked to the
external constraints imposed on the flow. This result is finally extended to a
heterogeneous maximal density constraint depending on time and space
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