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

We first develop non-oscillatory central schemes for a trafficflow 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 first-order staggered Lax-Friedrichs scheme and the second-order Nessyahu-Tadmor scheme, which belong to a class of Godunov-type projection-evolution 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 method, which is 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 a car's speed is affected more by nearby vehicles than distant (but still visible) ones. The Nessyahu-Tadmor scheme is 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.