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

Abstract

We introduce a model dealing with conservation laws on networks and coupled boundary conditions at the junctions. In particular, we introduce buffers of fixed arbitrary size and time dependent split ratios at the junctions, which represent how traffic is routed through the network, while guaranteeing spill-back phenomena at nodes. Having defined the dynamics at the level of conservation laws, we lift it up to the Hamilton-Jacobi (H-J) formulation and write boundary datum of incoming and outgoing junctions as functions of the queue sizes and vice-versa. The Hamilton-Jacobi formulation provides the necessary regularity estimates to derive a fixed-point problem in a proper Banach space setting, which is used to prove well-posedness of the model. Finally, we detail how to apply our framework to a non-trivial road network, with several intersections and finite-length links.

Burgers’ Equation Revisited: Extension of Mono-Dimensional Case on a Network

“…In the case of other vertices we may obtain the ambiguity. Consequently, imposing only conditions (17) on the non-negative weak solution to (16) still does not guarantee the uniqueness. Let us stop at this statement for a moment.…”

“…The problem of inviscid Burgers equation on networks belongs to the family of conservation laws on networks that has been developed for about thirty years and still receives considerable interest [5,12,23]. The major motivation for studying this topic is traffic modelling, see for instance [8,15,17], initiated with the well-established now Lighthill-Whitham model [20]. The natural interpretation of a graph as a transportation network shifted the burden of research interests into the case of non-convex flux which enforced the application of either wave-front tracking approximations or vanishing viscosity methods [7].…”

Branch Analysis of Macro-Traffic Flow Model With Different Driving Characteristics in Its Environment

“…The mathematical model describing the actual traffic law is the main tool to reveal the basic law of traffic flow. Researchers have now proposed a large number of traffic flow models, mainly including three types: cellular automata model [1][2][3] , micro car-following model [4][5][6][7][8] , and macro dynamics model [9][10][11][12][13] . However, due to the complexity of the traffic system, various traffic phenomena in actual traffic such as vehicles Stopping at all times, increased vehicle density, increased vehicle speed, etc.…”