Browse > Article
http://dx.doi.org/10.4134/JKMS.j190488

TRAFFIC FLOW MODELS WITH NONLOCAL LOOKING AHEAD-BEHIND DYNAMICS  

Lee, Yongki (Department of Mathematical Sciences Georgia Southern University)
Publication Information
Journal of the Korean Mathematical Society / v.57, no.4, 2020 , pp. 987-1004 More about this Journal
Abstract
Motivated by the traffic flow model with Arrhenius looka-head relaxation dynamics introduced in [25], this paper proposes a traffic flow model with look ahead relaxation-behind intensification by inserting look behind intensification dynamics to the flux. Finite time shock formation conditions in the proposed model with various types of interaction potentials are identified. Several numerical experiments are performed in order to demonstrate the performance of the modified model. It is observed that, comparing to other well-known macroscopic traffic flow models, the model equipped with look ahead relaxation-behind intensification has both enhanced dispersive and smoothing effects.
Keywords
Nonlocal conservation law; shock formation; traffic flow; global flux;
Citations & Related Records
연도 인용수 순위
  • Reference
1 S. A. Arrhenius, Uber die Dissociationswarme und den Einfluss der Temperatur auf den Dissociationsgrad der Elektrolyte, Z. Phys. Chem. 4 (1889), 96-116.   DOI
2 F. Betancourt, R. Burger, K. H. Karlsen, and E. M. Tory, On nonlocal conservation laws modelling sedimentation, Nonlinearity 24 (2011), no. 3, 855-885. https://doi.org/10.1088/0951-7715/24/3/008   DOI
3 M. Burger, Y. Dolak-Struss, and C. Schmeiser, Asymptotic analysis of an advection-dominated chemotaxis model in multiple spatial dimensions, Commun. Math. Sci. 6 (2008), no. 1, 1-28. http://projecteuclid.org/euclid.cms/1204905775   DOI
4 F. A. Chiarello and P. Goatin, Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel, ESAIM Math. Model. Numer. Anal. 52 (2018), no. 1, 163-180. https://doi.org/10.1051/m2an/2017066   DOI
5 A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181 (1998), no. 2, 229-243. https://doi.org/10.1007/BF02392586   DOI
6 Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity, SIAM J. Appl. Math. 66 (2005), no. 1, 286-308. https://doi.org/10.1137/040612841   DOI
7 S. Engelberg, H. Liu, and E. Tadmor, Critical thresholds in Euler-Poisson equations, Indiana Univ. Math. J. 50 (2001), Special Issue, 109-157. https://doi.org/10.1512/iumj.2001.50.2177   DOI
8 P. Goatin and S. Scialanga, Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity, Netw. Heterog. Media 11 (2016), no. 1, 107-121. https://doi.org/10.3934/nhm.2016.11.107   DOI
9 D. D. Holm and A. N. W. Hone, A class of equations with peakon and pulson solutions, J. Nonlinear Math. Phys. 12 (2005), suppl. 1, 380-394. https://doi.org/10.2991/jnmp. 2005.12.s1.31   DOI
10 J. K. Hunter, Numerical solutions of some nonlinear dispersive wave equations, in Computational solution of nonlinear systems of equations (Fort Collins, CO, 1988), 301-316, Lectures in Appl. Math., 26, Amer. Math. Soc., Providence, RI, 1990.
11 A. Keimer and L. Pflug, Existence, uniqueness and regularity results on nonlocal balance laws, J. Differential Equations 263 (2017), no. 7, 4023-4069. https://doi.org/10.1016/j.jde.2017.05.015   DOI
12 A. Keimer, L. Pflug, and M. Spinola, Nonlocal scalar conservation laws on bounded domains and applications in traffic flow, SIAM J. Math. Anal. 50 (2018), no. 6, 6271-6306. https://doi.org/10.1137/18M119817X   DOI
13 A. Kurganov and A. Polizzi, Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics, Netw. Heterog. Media 4 (2009), no. 3, 431-451. https://doi.org/10.3934/nhm.2009.4.431   DOI
14 G. Kynch, A theory of sedimentation, Trans. Fraday Soc, 48 (1952), 66-76.
15 M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London Ser. A 229 (1955), 317-345. https://doi.org/10.1098/rspa.1955.0089
16 Y. Lee and H. Liu, Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics, Discrete Contin. Dyn. Syst. 35 (2015), no. 1, 323-339. https://doi.org/10.3934/dcds.2015.35.323   DOI
17 D. Li and T. Li, Shock formation in a traffic flow model with Arrhenius look-ahead dynamics, Netw. Heterog. Media 6 (2011), no. 4, 681-694. https://doi.org/10.3934/nhm.2011.6.681   DOI
18 T. Li and H. Liu, Critical thresholds in hyperbolic relaxation systems, J. Differential Equations 247 (2009), no. 1, 33-48. https://doi.org/10.1016/j.jde.2009.03.032   DOI
19 H. Liu and E. Tadmor, Spectral dynamics of the velocity gradient field in restricted flows, Comm. Math. Phys. 228 (2002), no. 3, 435-466. https://doi.org/10.1007/s002200200667   DOI
20 H. Liu, Wave breaking in a class of nonlocal dispersive wave equations, J. Nonlinear Math. Phys. 13 (2006), no. 3, 441-466. https://doi.org/10.2991/jnmp.2006.13.3.8   DOI
21 P. I. Richards, Shock waves on the highway, Operations Res. 4 (1956), 42-51. https://doi.org/10.1287/opre.4.1.42   DOI
22 J. Rubinstein, Evolution equations for stratified dilute suspensions, Phys. Fluids A 2 (1990), no. 1, 3-6. https://doi.org/10.1063/1.857690   DOI
23 J. Rubinstein and J. B. Keller, Sedimentation of a dilute suspension, Phys. Fluids A 1 (1989), no. 4, 637-643. https://doi.org/10.1063/1.857438   DOI
24 R. Seliger, A note on the breaking of waves, Proc. Roy. Soc. Ser. A, 303 (1968), 493-496.
25 A. Sopasakis and M. A. Katsoulakis, Stochastic modeling and simulation of traffic flow: asymmetric single exclusion process with Arrhenius look-ahead dynamics, SIAM J. Appl. Math. 66 (2006), no. 3, 921-944. https://doi.org/10.1137/040617790   DOI
26 K. Zumbrun, On a nonlocal dispersive equation modeling particle suspensions, Quart. Appl. Math. 57 (1999), no. 3, 573-600. https://doi.org/10.1090/qam/1704419   DOI
27 E. Tadmor and C. Tan, Critical thresholds in flocking hydrodynamics with non-local alignment, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014), no. 2028, 20130401, 22 pp. https://doi.org/10.1098/rsta.2013.0401
28 V. O. Vakhnenko and E. J. Parkes, The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method, Chaos Solitons Fractals 13 (2002), no. 9, 1819-1826. https://doi.org/10.1016/S0960-0779(01)00200-4   DOI
29 G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1974.