Non-local traffic flow models with time delay: well-posedness and numerical approximation

TL;DR

The paper addresses a scalar non-local traffic flow equation with time delay, aiming to ensure well-posedness and physical admissibility. It introduces a non-local delayed conservation law with a saturation mechanism and analyzes two finite-volume schemes, Lax-Friedrichs and Hilliges-Weidlich, to construct approximate solutions. It proves uniform BV bounds, $\mathbf{L^1}$-stability with respect to initial data and delay, and existence of entropy weak solutions, along with convergence to the non-delayed model as $\tau \to 0$. Numerical experiments demonstrate the advantages of the HW scheme, the role of saturation in enforcing a maximum density, and the emergence of stop-and-go waves due to delay, with potential implications for multi-class and autonomous-vehicle–driven stabilization. Discrete compactness and entropy arguments underpin the results, making the framework suitable for future extensions and data-driven calibration.

Abstract

We prove the well-posedness of weak entropy solutions of a scalar non-local traffic flow model with time delay. Existence is obtained by convergence of finite volume approximate solutions constructed by Lax-Friedrich and Hilliges-Weidlich schemes, while the L1 stability with respect to the initial data and the delay parameter relies on a Kruzkov-type doubling of variable technique.Numerical tests are provided to illustrate the efficiency of the proposed schemes, as well as the solution dependence on the delay and look-ahead parameters.

Figures (8)

• Figure 1: Comparison between LF \ref{['schema']} and HW \ref{['schema2']} schemes for $\Delta x=5\cdot 10^{-3}$ corresponding to initial data \ref{['eq:ICtest1']}.
• Figure 2: Comparison between the solution to the model \ref{['delay_bis']} associated to the initial datum $\rho^0(x)=3/2\chi_{[1,2]}(x)$ and to the parameters $\tau=0.1$ and $L=0.15$, with decreasing values of the space step $\Delta x$.
• Figure 3: Comparison between the solution to the model with no saturation \ref{['delay']}, and the solution to the model \ref{['delay_bis']} with $\tau=0.12$, and with saturation functions \ref{['flinear']}\ref{['fexp']} . Two top rows: initial datum \ref{['eq:ICtest24']}; Two bottom rows: initial datum \ref{['eq:ICtest23']} with $\bar{\rho}=1/4$.
• Figure 4: Comparison between the solution to the model with no saturation \ref{['delay']}, and the solution to the model \ref{['delay_bis']} with saturation functions \ref{['flinear']}\ref{['fexp']}, associated to the initial datum \ref{['eq:ICtest23']} with $\bar{\rho}=1/2$ and to the delay $\tau=0.08$.
• Figure 5: Convergence of the delayed model \ref{['delay_bis']} to the non-delayed one \ref{['delay']} with velocity \ref{['Greennorm']}, linear decreasing kernel and exponential saturation function \ref{['fexp']} with $\varepsilon=1/50$, as the time delay parameter $\tau$ approaches zero. Left: initial datum \ref{['eq:ICtest21']}. Right: initial condition \ref{['datumnorm']}.

Theorems & Definitions (17)

• Remark 1
• Remark 2
• Definition 1: Entropy weak solution
• Lemma 1: Positivity
• Lemma 2: $\mathbf{L^1}$-bound
• Lemma 3: $\mathbf{L^\infty}$-bound / weak maximum principle
• Remark 3: Properties of the HW scheme
• Proposition 1: Spatial $\mathbf{BV}$-bound for the LF scheme
• Remark 4: Dependence on the parameters
• Proposition 2: Spatial $\mathbf{BV}$-bound for the HW scheme