Advanced

Developments in the theory of the Prigogine-Herman kinetic equation of vehicular traffic

Sopasakis, Alexandros LU (2000)
Abstract
The nonlinear kinetic equation of Prigogine and Herman is examined in regards to existence and uniqueness of solutions. The solution exists and is unique in the Banach space of bounded continuous functions over a particular subspace Ω.



The equilibrium solution of the kinetic equation of Prigogine and Herman is used to derive asymptotic type series expansions in the form of Hilbert or Chapman and Enskog for concentrations (c ) corresponding to the stable flow regime of traffic (0 < c < c crit ). As expected the conservation of mass equation, the Lighthill-Whitham-Richards model, can be obtained from these expansions.



We use the Chapman-Enskog expansion to obtain hydrodynamic-like equations... (More)
The nonlinear kinetic equation of Prigogine and Herman is examined in regards to existence and uniqueness of solutions. The solution exists and is unique in the Banach space of bounded continuous functions over a particular subspace Ω.



The equilibrium solution of the kinetic equation of Prigogine and Herman is used to derive asymptotic type series expansions in the form of Hilbert or Chapman and Enskog for concentrations (c ) corresponding to the stable flow regime of traffic (0 < c < c crit ). As expected the conservation of mass equation, the Lighthill-Whitham-Richards model, can be obtained from these expansions.



We use the Chapman-Enskog expansion to obtain hydrodynamic-like equations equivalent to the Euler, Navier-Stokes or Burnett equations of fluid flow, depending on the order of the series expansions we used. The zeroth and first order hydrodynamic-like partial differential equations are solved using appropriate conservative numerical schemes. Analogous continuum approximations up to order one are obtained from the Hilbert expansion.



Last a zeroth-order (extended Lighthill-Whitham-Richards) model is obtained for unstable flow at sufficiently high concentrations. (Less)
Please use this url to cite or link to this publication:
author
opponent
  • Prabir, Daripa, Mathematics, Texas A&M University
organization
publishing date
type
Thesis
publication status
published
subject
keywords
Chapman-Enskog expansion, Traffic, Prigogine-Herman, Kinetic equation, Unstable flow
pages
112 pages
defense location
College Station, Texas, USA
defense date
2000-05-26 10:00
ISBN
0-599-73811-1
978-0-599-73811-9
language
English
LU publication?
yes
id
92f33e0a-748a-4d71-9e43-21a4edd5e46d (old id 2201872)
date added to LUP
2016-03-14 14:42:00
date last changed
2016-09-19 08:45:18
@misc{92f33e0a-748a-4d71-9e43-21a4edd5e46d,
  abstract     = {The nonlinear kinetic equation of Prigogine and Herman is examined in regards to existence and uniqueness of solutions. The solution exists and is unique in the Banach space of bounded continuous functions over a particular subspace Ω.<br/><br>
<br/><br>
The equilibrium solution of the kinetic equation of Prigogine and Herman is used to derive asymptotic type series expansions in the form of Hilbert or Chapman and Enskog for concentrations (c ) corresponding to the stable flow regime of traffic (0 &lt; c &lt; c crit ). As expected the conservation of mass equation, the Lighthill-Whitham-Richards model, can be obtained from these expansions.<br/><br>
<br/><br>
We use the Chapman-Enskog expansion to obtain hydrodynamic-like equations equivalent to the Euler, Navier-Stokes or Burnett equations of fluid flow, depending on the order of the series expansions we used. The zeroth and first order hydrodynamic-like partial differential equations are solved using appropriate conservative numerical schemes. Analogous continuum approximations up to order one are obtained from the Hilbert expansion.<br/><br>
<br/><br>
Last a zeroth-order (extended Lighthill-Whitham-Richards) model is obtained for unstable flow at sufficiently high concentrations.},
  author       = {Sopasakis, Alexandros},
  isbn         = {0-599-73811-1},
  keyword      = {Chapman-Enskog expansion,Traffic,Prigogine-Herman,Kinetic equation,Unstable flow},
  language     = {eng},
  pages        = {112},
  title        = {Developments in the theory of the Prigogine-Herman kinetic equation of vehicular traffic},
  year         = {2000},
}