Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics

Abstract

We first develop non-oscillatory central schemes for a traffic flow model with Arrhenius look-ahead dynamics, proposed in [A. Sopasakis and M.A. Katsoulakis, SIAM J. Appl. Math., 66 (2006), pp. 921-944]. This model takes into account interactions of every vehicle with other vehicles ahead ("look-ahead" rule) and can be written as a one-dimensional scalar conservation law with a global flux. The proposed schemes are extensions of the nonoscillatory central schemes, which belong to a class of Godunov-type projectionevolution methods. In this framework, a solution, computed at a certain time, is first approximated by a piecewise polynomial function, which is then evolved to the next time level according to the integral form of the conservation law. Most Godunov-type schemes are based on upwinding, which requires solving (generalized) Riemann problems. However, no (approximate) Riemann problem solver is available for conservation laws with global fluxes. Therefore, central schemes, which are Riemann-problem-solver-free, are especially attractive for the studied traffic flow model. Our numerical experiments demonstrate high resolution, stability, and robustness of the proposed methods, which are used to numerically investigate both dispersive and smoothing effects of the global flux.We also modify the model by Sopasakis and Katsoulakis by introducing a more realistic, linear interaction potential that takes into account the fact that a car's speed is affected more by nearby vehicles than distant (but still visible) ones. The central schemes are extended to the modified model. Our numerical studies clearly suggest that in the case of a good visibility, the new model yields solutions that seem to better correspond to reality.

Threshold dynamics in hyperbolic partial differential equations

“…We should point out that compared to the recent work of T. Li and D. Li [Li et al (2011)], our shock formation conditions may be viewed in the perspective of critical thresholds. Fur-thermore, our shock formation conditions are consistent with the numerical results obtained in [Kurganov et al (2009)]. The results are more precisely stated in Chapter 4, together with relevant remarks.…”

“…The advection couples both local and nonlocal mechanisms. This class of conservation laws appears in several applications including traffic flows [Kurganov et al (2009); Sopasakis et al (2006)], the collective motion of biological cells [Dolak et al (2005); Burger et al (2008); Perthame et al (2009)], dispersive water waves [Whitham (1974);Holm et al (2005); Degasperis et al (1999); Liu (2006)], the radiating gas motion [Hamer (1971); Rosenau (1989); Liu et al (2001)], high-frequency waves in relaxing medium [Hunter (1990); Parkes (2002); Vakhnenko (1992)], and the kinematic sedimentation [Kynch (1952); Zumbrun The traffic flow model that motivated this study is the one with look-ahead relaxation introduced by Sopasakis and Katsoulakis [Sopasakis et al (2006)]:…”

“…This linear potential is intended to take into account the fact that a car's speed is affected more by nearby vehicles than distant ones. The authors in [Kurganov et al (2009)] carried out some careful numerical study of the traffic flow model (1.1.6), through three examples: red light traffic, traffic jam on a busy freeway and a numerical breakdown study. In the case of a good visibility (large γ), their numerical studies suggest that (1.1.6) with the modified potential (1.1.8) yields solutions that seem to better correspond to reality.…”

Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity

“…A finite volume central scheme. In this section, we describe a central scheme for (1) inspired by [18], which is an extension of the second-order Nessyahu-Tadmor scheme [22]. At each time step, the solution is first approximated by a piece-wise linear function, then evolved to the next time step according to the integral form of conservation law.…”

“…Formally, we have a second-order approximation provided the slopes s n j are at least first-order approximations of the derivatives ρ x (t n , x j ). We use a generalized minmod reconstruction (as in [18]) with:…”