558 research outputs found
From traffic and pedestrian follow-the-leader models with reaction time to first order convection-diffusion flow models
In this work, we derive first order continuum traffic flow models from a
microscopic delayed follow-the-leader model. Those are applicable in the
context of vehicular traffic flow as well as pedestrian traffic flow. The
microscopic model is based on an optimal velocity function and a reaction time
parameter. The corresponding macroscopic formulations in Eulerian or Lagrangian
coordinates result in first order convection-diffusion equations. More
precisely, the convection is described by the optimal velocity while the
diffusion term depends on the reaction time. A linear stability analysis for
homogeneous solutions of both continuous and discrete models are provided. The
conditions match the ones of the car-following model for specific values of the
space discretization. The behavior of the novel model is illustrated thanks to
numerical simulations. Transitions to collision-free self-sustained stop-and-go
dynamics are obtained if the reaction time is sufficiently large. The results
show that the dynamics of the microscopic model can be well captured by the
macroscopic equations. For non--zero reaction times we observe a scattered
fundamental diagram. The scattering width is compared to real pedestrian and
road traffic data
Properties of the LWR model with time delay
In this article, we investigate theoretical and numerical properties of the
first-order Lighthill-Whitham-Richards (LWR) traffic flow model with time
delay. Since standard results from the literature are not directly applicable
to the delayed model, we mainly focus on the numerical analysis of the proposed
finite difference discretization. The simulation results also show that the
delay model is able to capture Stop & Go waves
Traffic and Related Self-Driven Many-Particle Systems
Since the subject of traffic dynamics has captured the interest of
physicists, many astonishing effects have been revealed and explained. Some of
the questions now understood are the following: Why are vehicles sometimes
stopped by so-called ``phantom traffic jams'', although they all like to drive
fast? What are the mechanisms behind stop-and-go traffic? Why are there several
different kinds of congestion, and how are they related? Why do most traffic
jams occur considerably before the road capacity is reached? Can a temporary
reduction of the traffic volume cause a lasting traffic jam? Under which
conditions can speed limits speed up traffic? Why do pedestrians moving in
opposite directions normally organize in lanes, while similar systems are
``freezing by heating''? Why do self-organizing systems tend to reach an
optimal state? Why do panicking pedestrians produce dangerous deadlocks? All
these questions have been answered by applying and extending methods from
statistical physics and non-linear dynamics to self-driven many-particle
systems. This review article on traffic introduces (i) empirically data, facts,
and observations, (ii) the main approaches to pedestrian, highway, and city
traffic, (iii) microscopic (particle-based), mesoscopic (gas-kinetic), and
macroscopic (fluid-dynamic) models. Attention is also paid to the formulation
of a micro-macro link, to aspects of universality, and to other unifying
concepts like a general modelling framework for self-driven many-particle
systems, including spin systems. Subjects such as the optimization of traffic
flows and relations to biological or socio-economic systems such as bacterial
colonies, flocks of birds, panics, and stock market dynamics are discussed as
well.Comment: A shortened version of this article will appear in Reviews of Modern
Physics, an extended one as a book. The 63 figures were omitted because of
storage capacity. For related work see http://www.helbing.org
Gas-kinetic derivation of Navier-Stokes-like traffic equations
Macroscopic traffic models have recently been severely criticized to base on
lax analogies only and to have a number of deficiencies. Therefore, this paper
shows how to construct a logically consistent fluid-dynamic traffic model from
basic laws for the acceleration and interaction of vehicles. These
considerations lead to the gas-kinetic traffic equation of Paveri-Fontana. Its
stationary and spatially homogeneous solution implies equilibrium relations for
the `fundamental diagram', the variance-density relation, and other quantities
which are partly difficult to determine empirically.
Paveri-Fontana's traffic equation allows the derivation of macroscopic moment
equations which build a system of non-closed equations. This system can be
closed by the well proved method of Chapman and Enskog which leads to
Euler-like traffic equations in zeroth-order approximation and to
Navier-Stokes-like traffic equations in first-order approximation. The latter
are finally corrected for the finite space requirements of vehicles. It is
shown that the resulting model is able to withstand the above mentioned
criticism.Comment: For related work see
http://www.theo2.physik.uni-stuttgart.de/helbing.htm
Numerical Computation, Data Analysis and Software in Mathematics and Engineering
The present book contains 14 articles that were accepted for publication in the Special Issue “Numerical Computation, Data Analysis and Software in Mathematics and Engineering” of the MDPI journal Mathematics. The topics of these articles include the aspects of the meshless method, numerical simulation, mathematical models, deep learning and data analysis. Meshless methods, such as the improved element-free Galerkin method, the dimension-splitting, interpolating, moving, least-squares method, the dimension-splitting, generalized, interpolating, element-free Galerkin method and the improved interpolating, complex variable, element-free Galerkin method, are presented. Some complicated problems, such as tge cold roll-forming process, ceramsite compound insulation block, crack propagation and heavy-haul railway tunnel with defects, are numerically analyzed. Mathematical models, such as the lattice hydrodynamic model, extended car-following model and smart helmet-based PLS-BPNN error compensation model, are proposed. The use of the deep learning approach to predict the mechanical properties of single-network hydrogel is presented, and data analysis for land leasing is discussed. This book will be interesting and useful for those working in the meshless method, numerical simulation, mathematical model, deep learning and data analysis fields
- …