Second-order multilane traffic flow models: from the microscopic to the macroscopic scale

Abstract

This study addresses multilane vehicular traffic modelling, focusing on the rigorous transition from microscopic (individual vehicle-based) to macroscopic (aggregate flow-based) descriptions. While previous research on multilane traffic has largely focused on first-order models, we derive two novel multilane second-order macroscopic models by applying a microscopic-to-macroscopic limit to the multilane Bando-Follow-the-Leader model. These two models diverge fundamentally in their closure relations during the scaling limit: Model 1 directly projects the velocity dynamics, whereas Model 2 is built upon the heuristic conservation of the generalized momentum across lanes. Both models incorporate lane-changing dynamics through source terms in a hyperbolic system of balance laws, yet their structural differences lead to distinct relaxation limits, with Model 2 naturally relaxing to the first-order macroscopic multilane model recently derived by the authors, and Model 1 yielding a novel, non-standard first-order system. We propose several numerical experiments showing that the models can reproduce complex traffic phenomena, including congestion propagation, non-equilibrium effects, capacity drops, and asymmetric lane usage. Leveraging experimental datasets from real-world highways, we further construct lane-specific empirical fundamental diagrams and compare them with their simulated counterparts. The results demonstrate that the structural assumptions of Model 2 provide a superior quantitative fit with empirical data, faithfully capturing critical density values, traffic scattering, and characteristic lane-dependent patterns, thus offering a robust and generalizable tool for realistic traffic flow analysis.