Lagrangian-antidiffusive remap schemes for non-local multi-class traffic flow models

Abstract

This paper focuses on the numerical approximation of the solutions of a class of nonlocal systems in one space dimension, arising in traffic modeling. We propose alternative simple schemes by splitting the non-local conservation laws into two different equations, namely, the Lagrangian and the remap steps. We provide some properties and estimates recovered by approximating the problem with the Lagrangian-Antidiffusive Remap (L-AR) scheme, and we prove the convergence to weak solutions in the scalar case. Finally, we show some numerical simulations illustrating the efficiency of the L-AR schemes in comparison with classical first and second-order numerical schemes.From the application point of view, ρ i represents the density of the i-th vehicle class, characterized by its maximal velocity v max i and its interaction kernel ω i , which accounts for the reaction to downstream traffic distribution. In particular, η i is proportional to the look-ahead distance of drivers and J i represents the interaction strength. Equations (1.1) are coupled through the velocity function, which depends on an integral evaluation of the total traffic density r. For simplicity, in this work we will consider linear decreasing velocities setting ψ(r) = max{1 − r, 0}.Model (1.1) is a multi-class extension of the one-dimensional scalar conservation law with non-local flux proposed in [3,15]. It can also be seen as the non-local generalization of the "local" multi-population model for traffic flow described in [2]. Indeed, one of the limitations of the standard Lighthill-Whitham-Richards (LWR) traffic flow model [17,18] is the first in first out rule, conversely in multi-class dynamic faster vehicles can overtake slower ones and slower vehicles slow down the faster ones. In our setting, the non-local dependence of the speed functions v i describes the reaction of drivers that adapt their velocity with respect to what happens in front of them in terms of downstream traffic, assigning greater importance to closer vehicles, see also [9].Since solutions to (1.1), (1.4) may be discontinuous, they are intended in the following weak sense.

Stability of a Nonlocal Traffic Flow Model for Connected Vehicles

“…Let us also briefly compare this work with other relevant studies. The nonlocal LWR model (1.5) has been generalized to the case for 1-to-1 junctions [11] and of multi-class vehicles [13,15]. There are also nonlocal traffic flow models other than (1.5).…”

An Overview of Non-local Traffic Flow Models

“…Alternative first order schemes based on the splitting of the non-local equation in two equations, i.e. the Lagrangian and the remap steps are proposed in [14]. In comparison with Lax-Friedrichs scheme and upwind scheme, Lagrangian-Antidiffusive Remap schemes are much less diffusive.…”

“…In comparison with Lax-Friedrichs scheme and upwind scheme, Lagrangian-Antidiffusive Remap schemes are much less diffusive. Concerning high-order numerical schemes, it is worth citing the papers [8,13,18]. In [8], the authors propose discontinuous Galerkin and finite volume WENO schemes to obtain high-order approximations of non-local scalar conservation laws in one space dimension, where the velocity function depends on a weighted mean of the conserved quantity.…”

“…On the contrary, finite volume WENO schemes can be implemented on larger time steps. In [13], the authors extend to 1D systems the finite volume WENO schemes proposed in [8]. In [18], central WENO schemes are proposed, which, in contrast with the other high-order schemes for non-local conservation laws, neither require a restrictive CFL condition nor an additional reconstruction step.…”

A general result on the approximation of local conservation laws by nonlocal conservation laws: The singular limit problem for exponential kernels

“…Nonlocal conservation laws have been studied and analyzed quite intensively over the last decade from an application point of view with a particular focus on traffic flow [5,20,30,45,53,37], supply chains [39,55,32], pedestrian flow/crowd dynamics [21], opinion formation [2,51], chemical engineering [50,57], sedimentation [6], conveyor belts [54] and more. For the underlying dynamics existence and uniqueness [35,40,45,44,46,12,25], (optimal) control problems [33,4,14,19,38,24], and suitable numerical schemes [1,11,13,28,52] have been analyzed.…”