Adaptive numerical simulation of traffic flow density

In this paper, we describe a one-dimensional adaptive moving mesh method and apply it to the traffic flow model introduced in 2006 by Sopasakis and Katsoulakis and its modified model proposed in 2009 by Kurganov and Polizzi. These models can be written as a scalar conservation law with a global flux. The proposed scheme is an extension of the moving non-oscillatory central scheme, which belongs to a class of moving finite volume methods. We also modify the model given by Kurganov and Polizzi to account for the driver's reaction, i.e. delayed response. Finally, the moving finite volume method is extended accordingly.

[1]  Dirk Helbing,et al.  MASTER: macroscopic traffic simulation based on a gas-kinetic, non-local traffic model , 2001 .

[2]  Weizhang Huang,et al.  Moving mesh partial differential equations (MMPDES) based on the equidistribution principle , 1994 .

[3]  Tao Tang,et al.  Adaptive Mesh Methods for One- and Two-Dimensional Hyperbolic Conservation Laws , 2003, SIAM J. Numer. Anal..

[4]  Ali R. Soheili,et al.  Adaptive numerical method for Burgers-type nonlinear equations , 2012, Appl. Math. Comput..

[5]  Robert D. Russell,et al.  Adaptive Moving Mesh Methods , 2010 .

[6]  Riccardo Fazio,et al.  Moving-Mesh Methods for One-Dimensional Hyperbolic Problems Using CLAWPACK , 2003 .

[7]  Weizhang Huang,et al.  Analysis Of Moving Mesh Partial Differential Equations With Spatial Smoothing , 1997 .

[8]  H. M. Zhang,et al.  A Car-Following Theory for Multiphase Vehicular Traffic Flow , 2003 .

[9]  Arthur van Dam,et al.  A robust moving mesh finite volume method applied to 1D hyperbolic conservation laws from magnetohydrodynamics , 2006, J. Comput. Phys..

[10]  M J Lighthill,et al.  On kinematic waves II. A theory of traffic flow on long crowded roads , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[11]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[12]  Alexandros Sopasakis,et al.  Stochastic Modeling and Simulation of Traffic Flow: Asymmetric Single Exclusion Process with Arrhenius look-ahead dynamics , 2006, SIAM J. Appl. Math..

[13]  Alexandros Sopasakis.  Stochastic noise approach to traffic flow modeling , 2004 .

[14]  John M. Stockie,et al.  A moving mesh method with variable mesh relaxation time , 2008 .

[15]  Alexander Kurganov,et al.  Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics , 2009, Networks Heterog. Media.

[16]  E. Tadmor,et al.  Non-oscillatory central differencing for hyperbolic conservation laws , 1990 .

[17]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[18]  C. Chalons,et al.  Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling , 2008 .

[19]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[20]  John M. Stockie,et al.  A Moving Mesh Method for One-dimensional Hyperbolic Conservation Laws , 2000, SIAM J. Sci. Comput..

[21]  H. M. Zhang.  A finite difference approximation of a non-equilibrium traffic flow model , 2001 .

[22]  J. Hyman,et al.  An adaptive moving mesh method with static rezoning for partial differential equations , 2003 .

[23]  Carlos F. Daganzo,et al.  In Traffic Flow, Cellular Automata = Kinematic Waves , 2004 .