Abstract
We present a high-resolution relaxation scheme for a multi-class Lighthill-Whitham-Richards (MCLWR) traffic flow model. This scheme is based on high-order reconstruction for spatial discretization and an implicit-explicit Runge-Kutta method for time integration. The resulting method retains the simplicity of the relaxation schemes. There is no need to involve Riemann solvers and characteristic decomposition. Even the computation of the eigenvalues is not required. This makes the scheme particularly well suited for the MCLWR model in which the analytical expressions of the eigenvalues are difficult to obtain for more than four classes of road users. The numerical results illustrate the effectiveness of the presented method.
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References
Ascher, U.M., Ruuth, S.J., Spiteri, R.J., 1997. Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Appl. Numer. Math., 25(2–3):151–167. [doi:10.1016/S0168-9274(97)00056-1]
Banda, M., Seaïd, M., 2006. Non-oscillatory methods for relaxation approximation of Hamilton-Jacobi equations. Appl. Math. Comput., 183(1):170–1183. [doi:10.1016/j.amc.2006.05.066]
Banda, M., Seaïd, M., 2007. Relaxation WENO schemes for multidimensional hyperbolic systems of conservation laws. Numer. Methods Part. Diff. Equat., 23(5):1211–1234. [doi:10.1002/num.20218]
Banda, M.K., Seaïd, M., 2005. Higher-order relaxation schemes for hyperbolic systems of conservation laws. J. Numer. Math., 13(3):171–196. [doi:10.1515/156939505774286102]
Banda, M.K., Klar, A., Seaïd, M., 2007. A lattice-Boltzmann relaxation scheme for coupled convection-radiation systems. J. Comput. Phys., 226(2):1408–1431. [doi:10.1016/j.jcp.2007.05.030]
Chen, J.Z., Shi, Z.K., 2006. Application of a fourth-order relaxation scheme to hyperbolic systems of conservation laws. Acta Mech. Sin., 22(1):84–92. [doi:10.1007/s10409-005-0079-x]
Chen, J.Z., Shi, Z.K., Hu, Y.M., 2007. Numerical Solution of a Two-class LWR Traffic Flow Model by High-resolution Central-upwind Scheme. Proc. 7th Int. Conf. on Computational Science, p.17–24. [doi:10.1007/978-3-540-72584-8_3]
Drake, J.S., Schofer, J.L., May, A.D., 1967. A statistical analysis of speed density hypothesis. Highway Res. Rec., 154:53–87.
Jiang, R., Wu, Q.S., Zhu, Z.J., 2002. A new continuum model for traffic flow and numerical tests. Transp. Res. B, 36(5):405–419. [doi:10.1016/S0191-2615(01)00010-8]
Jin, S., Xin, Z., 1995. The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math., 48(3):235–276. [doi:10.1002/cpa.3160480303]
Kurganov, A., Nolle, S., Petrova, G., 2001. Semi-discrete central-upwind scheme for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput., 23(3):707–740. [doi:10.1137/S1064827500373413]
Lebacque, J.P., 1996. The Godunov Scheme and What It Means for First Order Traffic Flow Models. Proc. 13th Int. Symp. on Transportation and Traffic Theory, p.647–677.
LeVeque, R.J., 2002. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge, UK, p.378–388.
Levy, D., Puppo, G., Russo, G., 1999. Central WENO schemes for hyperbolic systems of conservation laws. ESAIM Math. Model. Numer. Anal., 33(3):547–571. [doi:10.1051/m2an:1999152]
Levy, D., Puppo, G., Russo, G., 2000. Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput., 22(2):656–672. [doi:10.1137/S1064827599359461]
Lighthill, M.J., Whitham, G.B., 1955. On kinematic waves (II): a theory of traffic flow on long crowed roads. Proc. R. Soc. Ser. A, 229(1178):317–345. [doi:10.1098/rspa.1955.0089]
Ngoduy, D., Liu, R., 2007. Multiclass first-order simulation model to explain non-linear traffic phenomena. Phys. A, 385(2):667–682. [doi:10.1016/j.physa.2007.07.041]
Richards, P.I., 1956. Shock waves on the highway. Oper. Res., 4(1):42–51. [doi:10.1287/opre.4.1.42]
Seaïd, M., 2004. Non-oscillatory relaxation methods for the shallow water equations in one and two space dimensions. Int. J. Numer. Methods Fluids, 46(5):457–484. [doi:10.1002/fld.766]
Shu, C.W., Osher, S., 1988. Effient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys., 77(2):439–471. [doi:10.1016/0021-9991(88)90177-5]
Wong, G.C.K., Wong, S.C., 2002. A multi-class traffic flow model: an extension of LWR model with heterogeneous drivers. Transp. Res. A, 36(9):827–841. [doi:10.1016/S0965-8564(01)00042-8]
Zhang, H.M., 2001. A finite difference approximation of a non-equilibrium traffic flow model. Transp. Res. B, 35(4):337–365. [doi:10.1016/S0191-2615(99)00054-5]
Zhang, H.M., 2002. A non-equilibrium traffic model devoid of gas-like behavior. Transp. Res. B, 36(3):275–290. [doi:10.1016/S0191-2615(00)00050-3]
Zhang, M.P., Shu, C.W., Wong, G.C.K., Wong, S.C., 2003. A weighted essentially non-oscillatory numerical scheme for a multi-class Lighthill-Whitham-Richards traffic flow model. J. Comput. Phys., 191(2):639–659. [doi:10.1016/S0021-9991(03)00344-9]
Zhang, P., Liu, R.X., Dai, S.Q., 2005. Theoretical analysis and numerical simulation on a two-phase traffic flow LWR model. J. Univ. Sci. Technol. China, 35(1):1–11 (in Chinese).
Zhang, P., Wong, S.C., Shu, C.W., 2006. A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway. J. Comput. Phys., 212(2):739–756. [doi:10.1016/j.jcp.2005.07.019]
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Project supported by the Aoxiang Project and the Scientific and Technological Innovation Foundation of Northwestern Polytechnical University, China (No. 2007KJ01011)
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Chen, Jz., Shi, Zk. & Hu, Ym. A relaxation scheme for a multi-class Lighthill-Whitham-Richards traffic flow model. J. Zhejiang Univ. Sci. A 10, 1835–1844 (2009). https://doi.org/10.1631/jzus.A0820829
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DOI: https://doi.org/10.1631/jzus.A0820829