Aw Rascle Research Articles

Bound-preserving OEDG schemes for Aw–Rascle–Zhang traffic models on networks

Physical solutions to the widely used Aw–Rascle–Zhang (ARZ) traffic model and the adapted pressure ARZ model should satisfy the positivity of density, the minimum and maximum principles with respect to the velocity v and other Riemann invariants. Many numerical schemes suffer from instabilities caused by violating these bounds, and the only existing bound-preserving (BP) numerical scheme (for ARZ model) is random, only first-order accurate, and not strictly conservative. This paper introduces arbitrarily high-order provably BP discontinuous Galerkin (DG) schemes for these two models, preserving all the aforementioned bounds except the maximum principle of v, which has been rigorously proven to conflict with the consistency and conservation of numerical schemes. Although the maximum principle of v is not directly enforced, we find that the strictly preserved maximum principle of another Riemann invariant w actually enforces an alternative upper bound on v. At the core of this work, analyzing and rigorously proving the BP property is a particularly nontrivial task: the Lax–Friedrichs (LF) splitting property, usually expected for hyperbolic conservation laws and employed to construct BP schemes, does not hold for these two models. To overcome this challenge, we formulate a generalized version of the LF splitting property, and prove it via the geometric quasilinearization approach (Wu and Shu, 2023 [47]). To suppress spurious oscillations in the DG solutions, we incorporate the oscillation-eliminating technique, recently proposed in (Peng et al., 2024 [34]), which is based on the solution operator of a novel damping equation. Several numerical examples are included to demonstrate the effectiveness, accuracy, and BP properties of our schemes, with applications to traffic simulations on road networks.

The hysteretic Aw–Rascle–Zhang model

AbstractA novel hyperbolic system of partial differential equations is introduced to model traffic flows. This system comprises three equations, with two being linearly degenerate; its main feature is the inclusion of a hysteretic term in a generalized Aw–Rascle–Zhang (ARZ) model. First, a maximum principle for the diffusive version of the model is proven. Then, it is demonstrated that a solution to the Riemann problem exists, which is unique among solutions that are monotone in velocity; all waves exploited in the construction have suitable viscous profiles. Through several examples it is shown how, as a consequence of different driving habits, the system can model the decay, emergence, or persistence of stop‐and‐go waves (a feature that is missing in the ARZ model), and such behavior is characterized by a simple geometric condition. Furthermore, the model allows the study of traffic flows with a mixture of drivers whose hysteresis loops are either clockwise or counterclockwise. In particular, the presence of sufficiently many of the former dampens speed oscillations.

Delta-shocks and vacuums in Riemann solutions to the Umami Chaplygin Aw–Rascle model

This paper is concerned with the formation of delta-shocks and vacuum states for zero-pressure Euler equations. With the introduction of the Umami Chaplygin gas, the Riemann problem for Aw–Rascle model with the reasonable flux-function is solved analytically. The three kinds of Riemann solutions involving the delta-shock are obtained. The generalized Rankine–Hugoniot relation and entropy condition for delta-shock are clarified. Under the entropy condition, the existence and uniqueness of the delta-shock solution are established by solving the generalized Rankine–Hugoniot relation. It is rigorously shown that as the Umami Chaplygin gas pressure and flux-function approach to zero simultaneously, the Riemann solutions of the considered model converge to these of the zero-pressure Euler equations. The numerical results support the theoretical analysis in the end.

A Game-Theoretic Framework for Generic Second-Order Traffic Flow Models Using Mean Field Games and Adversarial Inverse Reinforcement Learning

A traffic system can be interpreted as a multiagent system, wherein vehicles choose the most efficient driving approaches guided by interconnected goals or strategies. This paper aims to develop a family of mean field games (MFG) for generic second-order traffic flow models (GSOM), in which cars control individual velocity to optimize their objective functions. GSOMs do not generally assume that cars optimize self-interested objectives, so such a game-theoretic reinterpretation offers insights into the agents’ underlying behaviors. In general, an MFG allows one to model individuals on a microscopic level as rational utility-optimizing agents while translating rich microscopic behaviors to macroscopic models. Building on the MFG framework, we devise a new class of second-order traffic flow MFGs (i.e., GSOM-MFG), which control cars’ acceleration to ensure smooth velocity change. A fixed-point algorithm with fictitious play technique is developed to solve GSOM-MFG numerically. In numerical examples, different traffic patterns are presented under different cost functions. For real-world validation, we further use an inverse reinforcement learning approach (IRL) to uncover the underlying cost function on the next-generation simulation (NGSIM) data set. We formulate the problem of inferring cost functions as a min-max game and use an apprenticeship learning algorithm to solve for cost function coefficients. The results show that our proposed GSOM-MFG is a generic framework that can accommodate various cost functions. The Aw Rascle and Zhang (ARZ) and Light-Whitham-Richards (LWR) fundamental diagrams in traffic flow models belong to our GSOM-MFG when costs are specified. History: This paper has been accepted for the Transportation Science Special Issue on ISTTT25 Conference. Funding: X. Di is supported by the National Science Foundation [CAREER Award CMMI-1943998]. E. Iacomini is partially supported by the Italian Research Center on High-Performance Computing, Big Data and Quantum Computing (ICSC) funded by MUR Missione 4-Next Generation EU (NGEU) [Spoke 1 “FutureHPC & BigData”]. C. Segala and M. Herty thank the Deutsche Forschungsgemeinschaft (DFG) for financial support [Grants 320021702/GRK2326, 333849990/IRTG-2379, B04, B05, and B06 of 442047500/SFB1481, HE5386/18-1,19-2,22-1,23-1,25-1, ERS SFDdM035; Germany’s Excellence Strategy EXC-2023 Internet of Production 390621612; and Excellence Strategy of the Federal Government and the Länder]. Support through the EU DATAHYKING is also acknowledged. This work was also funded by the DFG [TRR 154, Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks, Projects C03 and C05, Project No. 239904186]. Moreover, E. Iacomini and C. Segala are members of the Indam GNCS (Italian National Group of Scientific Calculus).

Nonuniqueness of Weak Solutions to the Dissipative Aw–Rascle Model

We prove nonuniqueness of weak solutions to multi-dimensional generalisation of the Aw-Rascle model of vehicular traffic. Our generalisation includes the velocity offset in a form of gradient of density function, which results in a dissipation effect, similar to viscous dissipation in the compressible viscous fluid models. We show that despite this dissipation, the extension of the method of convex integration can be applied to generate infinitely many weak solutions connecting arbitrary initial and final states. We also show that for certain choice of data, ill posedness holds in the class of admissible weak solutions.

Adaptive observer design for coupled ODE–hyperbolic PDE systems with application to traffic flow estimation

This paper studies the state estimation problem for a class of coupled linear ODE–hyperbolic PDE systems with unknown in-domain parameters. Based on the swapping transformation and the least squares parameter estimation method, a Luenberger-type adaptive boundary observer is designed to estimate the ODE–PDE states under the unknown parameters. The exponential convergence conditions w.r.t. the observer feedback gains are given by employing the Lyapunov technique. Finally, an application to traffic state estimation of the Aw–Rascle–Zhang (ARZ) model involving the unknown relaxation time and the boundary input speed deviation is provided to validate the theoretical result.

A high-resolution meshfree particle method for numerical investigation of second-order macroscopic pedestrian flow models

Recent advancements in empirical observations of human psychological responses have revealed that the governing rules of human behavioral aspects can not only be predicted with adequate deterministic precision but also forged into mathematical formulations. These modeling studies enable crisis managers with reliable simulation tools to gain insights into critical facets of crowd disasters and enhance their decision-making abilities to facilitate safe egress in emergency situations. The macroscopic scale of representation in this context is often invoked to comprehend large-scale collective pattern formation in vast built environments, owing to its computational viability. The associated governing equations are typically constructed in a system of hyperbolic conservation law form, and conducting simulations in real-life complex geometric configurations can pose computational challenges. This article puts forward a high-resolution, shock-capturing meshfree particle method in an Eulerian framework for the numerical approximation of several widely adopted macroscopic pedestrian flow models. It serves as a superior alternative to traditional mesh-based methods, eliminating the need for specific mesh structures and connectivity between them. A classical Generalized Finite Difference Method (GFDM) in conjunction with several consistency conditions on spatial derivative coefficients is adopted to obtain a non-oscillatory and fairly conservative Godunov-type discretization of the governing system. The intercell flux is evaluated using a suitable Riemann solver, and a slope-limited reconstruction procedure of the corresponding Riemann states, adapted to arbitrary point distribution, is employed for improved accuracy. Moreover, the preferred direction of human motion is determined from a local density-dependent eikonal-type equation, which is solved using a generalized version of the Fast Marching Method. A thorough numerical investigation of three well-known second-order macroscopic models, namely Payne-Whitham (PW), Aw-Rascle (AR), and Aw-Rascle-Zhang (ARZ) is carried out through two hypothetical scenarios of crowd evacuation, not only to demonstrate the accuracy and applicability of the proposed approach but also to deepen understanding of their distinctive characteristics. A comparison of the flow flux with the fundamental diagram suggests that the density profile obtained with the ARZ model is equivalent to Hughes' first-order model. In contrast, the PW and AR models have proven to effectively replicate complex, unstable pedestrian phenomena such as stop-and-go waves. Finally, Helbing's counter-intuitive idea that strategically positioned barriers upstream of a bottleneck can essentially expedite evacuation is numerically investigated across various parameter choices.

Pedestrian models with congestion effects

We study the validity of the dissipative Aw–Rascle system as a macroscopic model for pedestrian dynamics. The model uses a congestion term (a singular diffusion term) to enforce capacity constraints in the crowd density while inducing a steering behavior. Furthermore, we introduce a semi-implicit, structure-preserving, and asymptotic-preserving numerical scheme which can handle the numerical solution of the model efficiently. We perform the first numerical simulations of the dissipative Aw–Rascle system in one and two dimensions. We demonstrate the efficiency of the scheme in solving an array of numerical experiments, and we validate the model, ultimately showing that it correctly captures the fundamental diagram of pedestrian flow.

Hard congestion limit of the dissipative Aw–Rascle system

In this study, we analyse the famous Aw–Rascle system in which the difference between the actual and the desired velocities (the offset function) is a gradient of a singular function of the density. This leads to a dissipation in the momentum equation which vanishes when the density is zero. The resulting system of PDEs can be used to model traffic or suspension flows in one dimension with the maximal packing constraint taken into account. After proving the global existence of smooth solutions, we study the so-called ‘hard congestion limit’, and show the convergence of a subsequence of solutions towards a weak solution of a hybrid free-congested system. This is also illustrated numerically using a numerical scheme proposed for the model studied. In the context of suspension flows, this limit can be seen as the transition from a suspension regime, driven by lubrication forces, towards a granular regime, driven by the contacts between the grains.

Delta shocks and vacuums in the Aw–Rascle model with anti van der Waals Chaplygin gas under the flux approximation

In this manuscript, we explore the concentration and cavitation phenomena in the Riemann problem for the Aw–Rascle model coupled with an anti van der Waals Chaplygin gas while considering a two-parameter flux approximation. We investigate the presence of a δ-shock and a vacuum state within the Riemann problem for this specific system. Additionally, we incorporate a perturbed flux approximation scheme and analyze the Riemann solution as the values of α1 and α2 approach 0. Our findings demonstrate that the δ-shock solution to the simplified equations can be achieved by examining the Riemann solution that involves two shock waves in the perturbed flux approximation system. This occurs when the flux approximation linked to the anti van der Waals Chaplygin gas model vanishes. Furthermore, the Riemann solution that includes two rarefaction waves converges to the vacuum state solution of the simplified equations.

Analysis of heterogeneous vehicular traffic: Using proportional densities

This paper presents a multi-class Aw–Rascle (AR) model with area occupancy expressed in terms of vehicle class proportions. Two vehicle classes that is; cars and motorcycles are considered based on an assumption that proportions of these form total traffic density. Qualitative properties of the proposed equilibrium velocity are established. Conditions under which the proposed model is stable are determined by linear stability analysis. To compute numerical flux, the model is decomposed by original Roe decomposition, where Roe matrix, averaged data variables and wave strengths are explicitly derived. The Roe matrix is shown to be hyperbolic, consistent and conservative. From the numerical results, the effect of proportional densities on the flow of vehicle classes is determined. Results obtained remain within limits therefore, the proposed model is realistic.

The perturbed Riemann problem for the Chaplygin pressure Aw–Rascle model with Coulomb-like friction

In this paper, we are concerned with the Riemann problem and the perturbed Riemann problem for the Chaplygin pressure Aw–Rascle model with Coulomb-like friction, which can also be seen as the nonsymmetric Keyfitz–Kranzer system with Chaplygin pressure and Coulomb-like friction. For the Riemann problem, we show that it explicitly exhibits two kinds of different structures and the delta shock wave appears in some certain situations. The generalized Rankine–Hugoniot conditions of the delta shock wave are established and the exact position, propagation speed and strength of the delta shock wave are given explicitly. Unlike the homogeneous case, it is shown that the Coulomb-like friction term makes contact discontinuities and the delta shock wave bend into parabolic shapes and the Riemann solutions are not self-similar anymore. For the perturbed Riemann problem with delta initial data, not only the delta shock wave but also the delta contact discontinuity are found in solutions and the friction term makes them bent. Under the generalized Rankine–Hugoniot conditions and the entropy condition, by taking variable substitution, we constructively obtain the global existence of generalized solutions which explicitly exhibit four kinds of different structures. The results in this paper yield a way of studying the wave interaction involving the delta shock wave for conservation laws with source terms and will give us some insights into the research on the Chaplygin pressure Aw–Rascle model, the pressureless Euler equations and the Chaplygin Euler equations with various kinds of source terms.

The transition of Riemann solutions with composite waves for the improved Aw–Rascle–Zhang model in dusty gas

We study the cavitation and concentration of the Riemann solutions for the improved Aw–Rascle–Zhang (IARZ) model in dusty gas with a non-genuinely nonlinear field. The Riemann solutions containing composite waves are constructed by Liu-entropy condition first. Second, we investigate the limits of the inflection point and tangent point along the 1-family wave curve and find that the composite waves tend to elementary waves as pressure vanishes. Third, we obtain the limiting behavior of the Riemann solutions and observe the formation of δ-shock wave and vacuum as pressure vanishes. We conclude that the limit of Riemann solutions of the IARZ model is not the Riemann solutions of the limit of the IARZ model. The phenomenon is consistent with the work of C. Shen and M. Sun [“Formation of delta-shocks and vacuum states in the vanishing pressure limit of solutions to the Aw–Rascle model,” J. Differ. Equations 249, 3024–3051 (2010)]. Finally, we perform some numerical simulations to verify our theoretical analysis.

Limits of solutions to the Aw-Rascle traffic flow model with generalized Chaplygin gas by flux approximation

The Riemann problem for the Aw-Rascle (AR) traffic flow model with a double parameter perturbation containing flux and generalized Chaplygin gas is first solved. Then, we show that the delta-shock solution of the perturbed AR model converges to that of the original AR model as the flux perturbation vanishes alone. Particularly, it is proved that as the flux perturbation and pressure decrease, the classical solution of the perturbed system involving a shock wave and a contact discontinuity will first converge to a critical delta shock wave of the perturbed system itself and only later to the delta-shock solution of the pressureless gas dynamics (PGD) model. This formation mechanism is interesting and innovative in the study of the AR model. By contrast, any solution containing a rarefaction wave and a contact discontinuity tends to a two-contact-discontinuity solution of the PGD model, and the nonvacuum intermediate state in between tends to a vacuum state. Finally, some representatively numerical results consistent with the theoretical analysis are presented.

Concentration of mass in the vanishing adiabatic exponent limit of Aw–Rascle traffic model with relaxation

Concentration of mass in the vanishing adiabatic exponent limit of Aw–Rascle traffic model with relaxation

Bifurcation Analysis of Improved Traffic Flow Model on Curved Road

Abstract Nonlinear analysis of complex traffic flow systems can provide a deep understanding of the causes of various traffic phenomena and reduce traffic congestion, and bifurcation analysis is a powerful method for it. In this paper, based on the improved Aw–Rascle model, a new macroscopic traffic flow model is proposed, which takes into account the road factors and driver psychological factor in the curve environment which can effectively simulate many realistic traffic phenomena on curves. The macroscopic traffic flow model on curved road is analyzed by bifurcation, first, it is transformed into a nonlinear dynamical system, then its stability conditions and the existence conditions of bifurcations are derived, and the changes of trajectories near the equilibrium points are described by phase plane. From an equilibrium point, various bifurcation structures describing the nonlinear traffic flow are obtained. In this paper, the influence of different bifurcations on traffic flow is analyzed, and the causes of special traffic phenomena such as stop-and-go and traffic clustering are described using Hopf bifurcation as the starting point of density temporal evolution. The derivation and simulation show that both road factors and driver psychological factor affect the stability of traffic flow on curves, and the study of bifurcation in the curved traffic flow model provides decision support for traffic management.

The Riemann problem for a traffic flow model

A traffic flow model describing the formation and dynamics of traffic jams was introduced by Berthelin et al. [“A model for the formation and evolution of traffic jams,” Arch. Ration. Mech. Anal. 187, 185–220 (2008)], which consists of a pressureless gas dynamics system under a maximal constraint on the density and can be derived from the Aw–Rascle model under the constraint condition ρ≤ρ* by letting the traffic pressure vanish. In this paper, we give up this constraint condition and consider the following form: {ρt+(ρu)x=0,(ρu+εp(ρ))t+(ρu2+εup(ρ))x=0,in which p(ρ)=−1ρ. The Riemann problem for the above traffic flow model is constructively solved. The delta shock wave arises in the Riemann solutions, although the system is strictly hyperbolic, its first eigenvalue is genuinely nonlinear, and the second eigenvalue is linearly degenerate. Furthermore, we clarify the generalized Rankine–Hugoniot relations and δ-entropy condition. The position, strength, and propagation speed of the delta shock wave are obtained from the generalized Rankine–Hugoniot conditions. The delta shock may be useful for the description of the serious traffic jam. More importantly, it is proved that the limits of the Riemann solutions of the above traffic flow model are exactly those of the pressureless gas dynamics system with the same Riemann initial data as the traffic pressure vanishes.

Riemann problem and wave interactions for an inhomogeneous Aw-Rascle traffic flow model with extended Chaplygin gas

The Riemann problem and wave interactions are discussed and investigated for an inhomogeneous Aw-Rascle (AR) traffic flow model with extended Chaplygin gas pressure. First, under some variable transformation, the Riemann problem with initial data of two piecewise constants is solved and two different types of Riemann solutions involving rarefaction wave, shock wave and contact discontinuity are obtained. Second, by studying the Riemann problem with three-piecewise-constant initial data, we analyze the interactions of waves and establish the global structures of Riemann solutions. It is shown that, influenced by the source term, the Riemann solutions for the inhomogeneous AR traffic flow model are no longer self-similar, and all the elementary wave curves do not keep straight. Finally, the stability of solution under the small perturbation of initial data is briefly discussed.

Lie Symmetry Analysis of the Aw–Rascle–Zhang Model for Traffic State Estimation

We extend our analysis on the Lie symmetries in fluid dynamics to the case of macroscopic traffic estimation models. In particular we study the Aw–Rascle–Zhang model for traffic estimation, which consists of two hyperbolic first-order partial differential equations. The Lie symmetries, the one-dimensional optimal system and the corresponding Lie invariants are determined. Specifically, we find that the admitted Lie symmetries form the four-dimensional Lie algebra A4,12. The resulting one-dimensional optimal system is consisted by seven one-dimensional Lie algebras. Finally, we apply the Lie symmetries in order to define similarity transformations and derive new analytic solutions for the traffic model. The qualitative behaviour of the solutions is discussed.

Controller design for heterogeneous traffic with bottleneck and disturbances

This paper studies an optimal tuning of the boundary controller for a heterogeneous traffic flow model with disturbances in order to alleviate congested traffic. The macroscopic first-order N-class Aw–Rascle traffic model consists of 2N hyperbolic partial differential equations. The vehicle size and the driver’s behavior characterize the type of vehicle. There are m positive characteristic velocities and 2N−m negative characteristic velocities in the congested traffic after linearizing the model equations around the steady state depending on the spatial variable. By using the backstepping method, a controller implemented by a ramp metering at the inlet boundary is designed for rejecting the disturbances to stabilize the 2N×2N heterogeneous traffic system. The developed controller in terms of proportional integral control is derived from mapping the original system to a target system with a proportional integral boundary control rejecting the disturbances. The integral input-to-state stability of the target system is proved by using the Lyapunov method. Finally, an optimization problem is established and solved for seeking the optimal tuning of the controller.