A non-local system modeling bi-directional traffic flows (Q6613476)

The authors investigate a mathematical model addressing the dynamics of two groups of agents moving in opposite directions. This research contributes to the macroscopic traffic flow literature by introducing a system of conservation laws with non-local fluxes, which are coupled in the speed function.\N\NThe scientific problem addressed in the paper lies in modelling and analysing bi-directional traffic flows using a macroscopic framework. Traditional traffic flow models such as the Lighthill-Whitham-Richards (LWR) model focus on local dynamics. In contrast, the authors extend the classical model by incorporating non-local effects, allowing the velocity of agents to depend on a weighted average of downstream traffic density. This approach accounts for the behavioural adaptation of agents based on their observations of traffic conditions ahead. The study draws from a mixed-type system proposed in earlier works, where hyperbolicity and the existence of solutions were challenging to establish due to oscillations arising in non-hyperbolic regions of the phase space. By introducing non-local terms in the speed functions, the authors aim to ensure the existence of solutions while preserving the physical realism of the model.\N\NTo address these challenges, the authors formulate a \(2 \times 2\) system of conservation laws with non-local fluxes. The density of agents moving in each direction is governed by equations that incorporate convolution terms representing the weighted densities of both groups within specified visibility ranges. The kernels characterising these convolutions are designed to respect physical constraints such as positivity, normalisation, and compact support. The authors employ a finite volume numerical scheme to approximate solutions, ensuring convergence under a suitable CFL condition. Rigorous analysis establishes key properties of the approximate solutions, including positivity, boundedness in \( L^1 \) and \( L^\infty \), and bounded total variation. These properties form the basis for proving the existence of weak solutions locally in time using a Lax-Wendroff type argument and Helly's theorem.\N\NThe study presents several numerical tests to explore the behaviour of the proposed model under varying conditions. These tests examine the impact of kernel support size, periodic boundary conditions, and agent heterogeneity in velocity and look-ahead distance. The results demonstrate that the solutions of the non-local model exhibit consistent behaviour with their local counterparts when kernel supports are sufficiently small. Furthermore, the numerical simulations reveal the role of kernel and velocity parameters in influencing solution stability and oscillatory behaviour. For example, agents with longer look-ahead distances exhibit reduced oscillations, highlighting the importance of non-local terms in regulating traffic dynamics.\N\NThis research is significant for its theoretical and practical contributions to the field of traffic flow modelling. The incorporation of non-local effects addresses limitations of classical models by capturing more realistic agent interactions. The authors' analytical framework ensures mathematical rigour, while the numerical experiments provide insights into practical scenarios, paving the way for applications in traffic management and urban planning. By bridging theoretical developments and empirical observations, this work offers a robust foundation for further investigations into non-local macroscopic models across diverse fields.\N\NFor the entire collection see [Zbl 1530.65011].