16 research outputs found
Lane formation by side-stepping
In this paper we study a system of nonlinear partial differential equations,
which describes the evolution of two pedestrian groups moving in opposite
direction. The pedestrian dynamics are driven by aversion and cohesion, i.e.
the tendency to follow individuals from the own group and step aside in the
case of contraflow. We start with a 2D lattice based approach, in which the
transition rates reflect the described dynamics, and derive the corresponding
PDE system by formally passing to the limit in the spatial and temporal
discretization. We discuss the existence of special stationary solutions, which
correspond to the formation of directional lanes and prove existence of global
in time bounded weak solutions. The proof is based on an approximation argument
and entropy inequalities. Furthermore we illustrate the behavior of the system
with numerical simulations
Cross-diffusion systems with excluded volume effects and asymptotic gradient flows
In this paper we discuss the analysis of a cross-diffusion PDE system for a
mixture of hard spheres, which was derived by Bruna and Chapman from a
stochastic system of interacting Brownian particles using the method of matched
asymptotic expansions. The resulting cross-diffusion system is valid in the
limit of small volume fraction of particles. While the system has a gradient
flow structure in the symmetric case of all particles having the same size and
diffusivity, this is not valid in general. We discuss local stability and
global existence for the symmetric case using the gradient flow structure and
entropy variable techniques. For the general case, we introduce the concept of
an asymptotic gradient flow structure and show how it can be used to study the
behavior close to equilibrium. Finally we illustrate the behavior of the model
with various numerical simulations
From nonlocal to local Cahn-Hilliard equation
In this paper we prove the convergence of a nonlocal version of the
Cahn-Hilliard equation to its local counterpart as the nonlocal convolution
kernel is scaled using suitable approximations of a Dirac delta in a periodic
boundary conditions setting. This convergence result strongly relies on the
dynamics of the problem. More precisely, the -gradient flow structure
of the equation allows to deduce uniform estimates for solutions of the
nonlocal Cahn-Hilliard equation and, together with a Poincar\'e type inequality
by Ponce, provides the compactness argument that allows to prove the
convergence result.Comment: 13 page
Numerical simulation of continuity equations by evolving diffeomorphisms
In this paper we present a numerical scheme for nonlinear continuity equations, which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance. It can be applied to a large class of nonlinear continuity equations, whose dynamics are driven by internal energies, given external potentials and/or interaction energies. The solver is based on its variational formulation as a gradient flow with respect to the Wasserstein distance. Positivity of solutions as well as energy decrease of the semi-discrete scheme are guaranteed by its construction. We illustrate this properties with various examples in spatial dimension one and two
Derivation and analysis of continuum models for crossing pedestrian traffic
In this paper we study hyperbolic and parabolic nonlinear partial differential equation models, which describe the evolution of two intersecting pedestrian flows. We assume that individuals avoid collisions by sidestepping, which is encoded in the transition rates of the microscopic 2D model. We formally derive the corresponding mean-field models and prove existence of global weak solutions for the parabolic model. Moreover we discuss stability of stationary states for the corresponding one-dimensional model. Furthermore we illustrate the rich dynamics of both systems with numerical simulations