[Paper Review] On the Existence of Solution of Conservation Law with Moving Bottleneck and Discontinuity in FLux

Hossein Nick Zinat Matin, Maria Laura Delle Monache|arXiv (Cornell University)|Oct 1, 2023

Navier-Stokes equation solutions4 citations

TL;DR

This paper establishes the existence of a weak solution for a PDE-ODE model of traffic flow with a moving bottleneck and discontinuous flux using a wavefront tracking scheme. By introducing a modified Riemann solution and analyzing total variation via a homeomorphism, the authors prove convergence of approximate solutions despite resonance and loss of uniform total variation due to flux discontinuities and flux constraints.

ABSTRACT

In this paper, a PDE-ODE model with discontinuity in the flux as well as a flux constraint is analyzed. A modified Riemann solution is proposed and the existence of a weak solution to the Cauchy problem is rigorously investigated using the wavefront tracking scheme.

Motivation & Objective

• To rigorously investigate the existence of weak solutions for a conservation law with discontinuous flux and flux constraints in traffic flow.
• To extend prior PDE-ODE models by incorporating moving bottlenecks and discontinuities in the flux function.
• To address the challenge of non-strict hyperbolicity and loss of uniform total variation due to flux discontinuities.
• To validate the convergence of approximate solutions using wavefront tracking and a homeomorphism-based total variation analysis.
• To generalize existing results on moving bottlenecks by including discontinuous flux functions, relevant to variable speed limits or road segment changes.

Proposed method

• A modified Riemann solution is constructed to handle discontinuities in the flux function, particularly at points where the flux changes abruptly.
• The wavefront tracking scheme is employed to construct approximate solutions, tracking the evolution of shocks and rarefactions.
• A homeomorphism is used to analyze the total variation of the flux function, enabling compactness arguments despite loss of uniform total variation.
• The flux constraint is modeled as a capacity reduction in the Lighthill-Whitham-Richards (LWR) model, coupled with an ODE for the bottleneck's trajectory.
• The convergence of the approximate solutions to a weak solution is proven by establishing bounded total variation and compactness in the transformed space.
• A singular map technique, inspired by Temple’s approach, is applied to handle resonance and ensure convergence in the presence of discontinuous flux.

Experimental results

Research questions

• RQ1 Does a weak solution exist for a PDE-ODE model of traffic flow with a moving bottleneck and discontinuous flux?
• RQ2 How can the wavefront tracking scheme be adapted to handle flux discontinuities and flux constraints simultaneously?
• RQ3 What conditions ensure the convergence of approximate solutions when the flux function is discontinuous and the system is resonant?
• RQ4 Can the total variation be controlled in the presence of discontinuous flux and non-classical waves?
• RQ5 How does the inclusion of a moving bottleneck affect the structure and stability of solutions in a discontinuous flux setting?

Key findings

• The existence of a weak solution to the Cauchy problem is rigorously established for the PDE-ODE model with discontinuous flux and flux constraint.
• A modified Riemann solution is introduced to handle the interaction between discontinuous flux and moving bottlenecks, ensuring consistency at jump points.
• The wavefront tracking scheme is adapted to maintain bounded total variation through a homeomorphism, enabling convergence despite loss of uniform total variation.
• The minimum distance between two densities in the solution is bounded below by a function of time and the distance to the next grid point in the velocity space, ensuring wave separation.
• The convergence of the approximate solutions to a weak solution is proven using compactness and bounded variation in the transformed space, even under resonance conditions.
• The method generalizes prior results by incorporating both moving bottlenecks and discontinuous flux, a novel combination in the literature.