DOI QR코드

DOI QR Code

TRAFFIC FLOW MODELS WITH NONLOCAL LOOKING AHEAD-BEHIND DYNAMICS

  • Lee, Yongki (Department of Mathematical Sciences Georgia Southern University)
  • 투고 : 2019.07.11
  • 심사 : 2019.12.12
  • 발행 : 2020.07.01

초록

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.

키워드

참고문헌

  1. S. A. Arrhenius, Uber die Dissociationswarme und den Einfluss der Temperatur auf den Dissociationsgrad der Elektrolyte, Z. Phys. Chem. 4 (1889), 96-116. https://doi.org/10.1515/zpch-1889-0408
  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
  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 https://doi.org/10.4310/CMS.2008.v6.n1.a1
  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
  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
  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
  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
  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
  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
  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
  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
  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
  14. G. Kynch, A theory of sedimentation, Trans. Fraday Soc, 48 (1952), 66-76.
  15. 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
  16. 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
  17. 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
  18. 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
  19. 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
  20. 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
  21. P. I. Richards, Shock waves on the highway, Operations Res. 4 (1956), 42-51. https://doi.org/10.1287/opre.4.1.42
  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
  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
  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
  26. 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
  27. 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
  28. G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1974.
  29. 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