Convergence of a misanthrope process to the entropy solution of 1D problems

We prove the convergence, in some strong sense, of a Markov process called “a misanthrope process” to the entropy weak solution of a one-dimensional scalar nonlinear hyperbolic equation. Such a process may be used for the simulation of traffic flows. The convergence proof relies on the uniqueness of entropy Young measure solutions to the nonlinear hyperbolic equation, which holds for both the bounded and the unbounded cases. In the unbounded case, we also prove an error estimate. Finally, numerical results show how this convergence result may be understood in practical cases.

[1]  Kerner,et al.  Experimental properties of complexity in traffic flow. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[2]  Benoît Merlet,et al.  Error estimate for finite volume scheme , 2007, Numerische Mathematik.

[3]  Sophie Mercier,et al.  An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic Markov processes , 2008 .

[4]  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..

[5]  Fraydoum Rezakhanlou,et al.  Hydrodynamic limit for attractive particle systems on 417-1417-1417-1 , 1991 .

[6]  Julien Vovelle,et al.  Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains , 2002, Numerische Mathematik.

[7]  R. J. Diperna,et al.  Measure-valued solutions to conservation laws , 1985 .

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

[9]  Antoine Tordeux.  Étude de processus en temps continu modélisant l'écoulement de flux de trafic routier , 2010 .

[10]  Long Chen.  FINITE VOLUME METHODS , 2011 .

[11]  M. Vares,et al.  Hydrodynamic equations for attractive particle systems on ℤ , 1987 .

[12]  D. Helbing.  Traffic and related self-driven many-particle systems , 2000, cond-mat/0012229.

[13]  F. Rezakhanlou.  Hydrodynamic limit for attractive particle systems on ${\bf Z}^d$ , 1991 .

[14]  J. Nédélec,et al.  First order quasilinear equations with boundary conditions , 1979 .

[15]  S. Kružkov.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES , 1970 .

[16]  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.

[17]  Masahiro Kanai,et al.  Two-lane traffic-flow model with an exact steady-state solution. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  C. Cocozza-Thivent,et al.  Quelques compléments sur le processus des misanthropes et le processus «zéro-range» , 1985 .

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

[20]  K. Ravishankar,et al.  Euler hydrodynamics of one-dimensional attractive particle systems , 2006 .

[21]  R Mahnke,et al.  Zero-range model of traffic flow. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Christiane Cocozza-Thivent,et al.  Processus des misanthropes , 1985 .

[23]  Raimund Bürger,et al.  The initial-boundary value problem for a scalar conservation law , 1999 .

[24]  F. Rezakhanlou.  Hydrodynamic limit for attractive particle systems on Zd , 1991 .

[25]  T. Gobron,et al.  Couplings, attractiveness and hydrodynamics for conservative particle systems , 2009, 0903.0316.

[26]  Thierry Gallouët,et al.  Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh , 1993 .

[27]  C. Daganzo.  THE CELL TRANSMISSION MODEL.. , 1994 .

[28]  W. Y. Szeto,et al.  Stochastic cell transmission model (SCTM): A stochastic dynamic traffic model for traffic state surveillance and assignment , 2011 .

[29]  F. Spitzer.  Interaction of Markov processes , 1970 .

[30]  R. Eymard,et al.  Limit boundary conditions for finite volume approximations of some physical problems , 2003 .

[31]  C. Landim,et al.  Scaling Limits of Interacting Particle Systems , 1998 .

[32]  P. I. Richards.  Shock Waves on the Highway , 1956 .