A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function

The well-known Lighthill-Whitham-Richards (LWR) kinematic model of traffic flow models the evolution of the local density of cars by a nonlinear scalar conservation law. The transition between free and congested flow regimes can be described by a flux or velocity function that has a discontinuity at a determined density. A numerical scheme to handle the resulting LWR model with discontinuous velocity was proposed in [J.D. Towers, A splitting algorithm for LWR traffic models with flux discontinuities in the unknown, J. Comput. Phys., 421 (2020), article 109722]. A similar scheme is constructed by decomposing the discontinuous velocity function into a Lipschitz continuous function plus a Heaviside function and designing a corresponding splitting scheme. The part of the scheme related to the discontinuous flux is handled by a semi-implicit step that does, however, not involve the solution of systems of linear or nonlinear equations. It is proved that the whole scheme converges to a weak solution in the scalar case. The scheme can in a straightforward manner be extended to the multiclass LWR (MCLWR) model, which is defined by a hyperbolic system of \begin{document}$ N $\end{document} conservation laws for \begin{document}$ N $\end{document} driver classes that are distinguished by their preferential velocities. It is shown that the multiclass scheme satisfies an invariant region principle, that is, all densities are nonnegative and their sum does not exceed a maximum value. In the scalar and multiclass cases no flux regularization or Riemann solver is involved, and the CFL condition is not more restrictive than for an explicit scheme for the continuous part of the flux. Numerical tests for the scalar and multiclass cases are presented.

[1]  Pep Mulet,et al.  A secular equation for the Jacobian matrix of certain multispecies kinematic flow models , 2010 .

[2]  Pep Mulet,et al.  Characteristic-Based Schemes for Multi-Class Lighthill-Whitham-Richards Traffic Models , 2008, J. Sci. Comput..

[3]  J. Dias,et al.  On the Riemann problem for some discontinuous systems of conservation laws describing phase transitions , 2004 .

[4]  N. Risebro,et al.  Solution of the Cauchy problem for a conservation law with a discontinuous flux function , 1992 .

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

[6]  S. Wong,et al.  Hyperbolicity and kinematic waves of a class of multi-population partial differential equations , 2006, European Journal of Applied Mathematics.

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

[8]  P. Gwiazda,et al.  MULTI-DIMENSIONAL SCALAR CONSERVATION LAWS WITH FLUXES DISCONTINUOUS IN THE UNKNOWN AND THE SPATIAL VARIABLE , 2013 .

[9]  T. Gimse Conservation laws with discontinuous flux functions , 1993 .

[10]  J. Dias,et al.  On the approximation of the solutions of the Riemann problem for a discontinuous conservation law , 2005 .

[11]  L. M. Villada,et al.  A Diffusively Corrected Multiclass Lighthill-Whitham-Richards Traffic Model with Anticipation Lengths and Reaction Times , 2013 .

[12]  Yadong Lu,et al.  The Entropy Solutions for the Lighthill-Whitham-Richards Traffic Flow Model with a Discontinuous Flow-Density Relationship , 2009, Transp. Sci..

[13]  Shing Chung Josh Wong,et al.  A multi-class traffic flow model: an extension of LWR model with heterogeneous drivers , 2002 .

[14]  Stefan Diehl,et al.  A conservation Law with Point Source and Discontinuous Flux Function Modelling Continuous Sedimentation , 1996, SIAM J. Appl. Math..

[15]  John D. Towers A splitting algorithm for LWR traffic models with flux discontinuous in the unknown , 2020, J. Comput. Phys..

[16]  S. Wong,et al.  A note on the weighted essentially non‐oscillatory numerical scheme for a multi‐class Lighthill–Whitham–Richards traffic flow model , 2009 .

[17]  Sze Chun Wong,et al.  A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway , 2006, J. Comput. Phys..

[18]  Raimund Bürger,et al.  Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model , 2008, Networks Heterog. Media.

[19]  R. Colombo,et al.  Measure valued solutions to conservation laws motivated by traffic modelling , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[20]  John M. Stockie,et al.  Riemann solver for a kinematic wave traffic model with discontinuous flux , 2011, J. Comput. Phys..

[21]  Piotr Gwiazda,et al.  ON SCALAR HYPERBOLIC CONSERVATION LAWS WITH A DISCONTINUOUS FLUX , 2011 .

[22]  Raimund Bürger,et al.  On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux , 2010, Networks Heterog. Media.

[23]  J. Vovelle,et al.  CONVERGENCE OF IMPLICIT FINITE VOLUME METHODS FOR SCALAR CONSERVATION LAWS WITH DISCONTINUOUS FLUX FUNCTION , 2008 .

[24]  Raimund Bürger,et al.  Second-order schemes for conservation laws with discontinuous flux modelling clarifier–thickener units , 2010, Numerische Mathematik.

[25]  Rinaldo M. Colombo,et al.  Hyperbolic Phase Transitions in Traffic Flow , 2003, SIAM J. Appl. Math..

[26]  J. Rodrigues,et al.  Solutions to a Scalar Discontinuous Conservation Law in a Limit Case of Phase Transitions , 2005 .

[27]  Chi-Wang Shu,et al.  A weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill-Whitham-Richards traffic flow model , 2003 .

[28]  Martin Hilliges,et al.  A phenomenological model for dynamic traffic flow in networks , 1995 .

[29]  Raimund Bürger,et al.  Antidiffusive and Random-Sampling Lagrangian-Remap Schemes for the Multiclass Lighthill-Whitham-Richards Traffic Model , 2013, SIAM J. Sci. Comput..

[30]  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.

[31]  Rinaldo M. Colombo,et al.  An $n$-populations model for traffic flow , 2003, European Journal of Applied Mathematics.

[32]  J. Carrillo Conservation laws with discontinuous flux functions and boundary condition , 2003 .

[33]  Raimund Bürger,et al.  A family of numerical schemes for kinematic flows with discontinuous flux , 2008 .

[34]  Shing Chung Josh Wong,et al.  A hybrid scheme for solving a multi-class traffic flow model with complex wave breaking , 2008 .