Developments in the theory of the Prigogine-Herman kinetic equation of vehicular traffic
Author
Summary, in English
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.
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.
Department/s
- Mathematics (Faculty of Engineering)
- Partial differential equations
Publishing year
2000
Language
English
Document type
Dissertation
Topic
- Mathematics
Keywords
- Chapman-Enskog expansion
- Traffic
- Prigogine-Herman
- Kinetic equation
- Unstable flow
Status
Published
Research group
- Partial differential equations
Supervisor
- Paul Nelson
ISBN/ISSN/Other
- ISBN: 0-599-73811-1
- ISBN: 978-0-599-73811-9
Defence date
26 May 2000
Defence time
10:00
Defence place
College Station, Texas, USA
Opponent
- Daripa Prabir