The Riemann solver for traffic flow at an intersection with buffer of vanishing size

Abstract

The paper examines the model of traffic flow at an intersection introduced in [2], containing a buffer with limited size. As the size of the buffer approach zero, it is proved that the solution of the Riemann problem with buffer converges to a self-similar solution described by a specific Limit Riemann Solver (LRS). Remarkably, this new Riemann Solver depends Lipschitz continuously on all parameters.

A macroscopic traffic flow model with finite buffers on networks: well-posedness by means of Hamilton–Jacobi equations

“…Assumption 2.1 and also the regularity proposed for defining the Legendre-Fenchel transform in Section 4.1 can be weakened to f ∈ C 1 (R), strictly concave with super-linear growth. However, in the framework of traffic flow modelling, the additional regularity f ∈ C 2 (R) is not too restrictive and is aligned with [8][9][10], on which this contribution is based on.…”

“…Hamilton-Jacobi equations on networks without buffers were developed in [1,30]. Our analysis strongly relies on previous contributions in [8][9][10], where a similar fixed-point problem was posed. In these articles, the authors assume that the routing of each population is predetermined initially, and use transport equations to propagate this information and to keep track of different populations having different routes.…”

“…The state of this buffer is then governed by an ordinary differential equation, taking into account the boundary conditions at the junction so that conservation of mass is guaranteed. This modeling framework was first introduced for supply-chain networks in [4,25,26,28] and then adapted to traffic flow on networks in [8][9][10]20,24,29]. This approach provides stability estimates which are crucial from a control point of view, but may lead to a potential loss of information at the junctions, depending on how the buffer is modeled and whether one aggregates commodities at the junction level or not.…”

Entropy Dissipation at the Junction for Macroscopic Traffic Flow Models

“…The LWR model on networks and different coupling conditions were studied by many authors, see e.g. [3,11,13,14,17,18,22,29]. For a general introduction to traffic flow on networks, see [19,20].…”

On the Optimization of Conservation Law Models at a Junction with Inflow and Flow Distribution Controls

“…In this model the unique solution of a general Cauchy problem (1.2)-(1.3) is obtained employing an extension of the Lax-Oleǐnik formula to the initial boundary value problem and determining the length of queues inside a buffer as the fixed point of a contractive transformation. In fact, it is shown in [18] that a specific Riemann Solver at the junction determines the limiting solution obtained in presence of a buffer when the buffer's capacity approaches zero. Instead, a Junction Riemann Solver for a source destination model that incorporates a dynamic description of the car path choices was previously introduced in [40].…”