A Nonlinear Heat Equation Arising from Automated-Vehicle Traffic Flow Models

TL;DR

This paper introduces a new nonlinear heat equation modeling automated vehicle traffic, analyzes its solutions using entropy conditions, and proposes a conservative finite difference scheme validated through traffic simulations.

Contribution

It presents a novel nonlinear heat equation for automated vehicle traffic, develops an entropy-respecting numerical scheme, and compares it with existing models through simulations.

Findings

The proposed scheme accurately approximates the weak solution.

Simulations demonstrate benefits of automated vehicles over traditional models.

The entropy conditions ensure physically relevant solutions.

Abstract

In this paper, a new nonlinear heat equation is studied that arises as a model of the collective behavior of automated vehicles. The properties of the solutions of this equation are studied by introducing the appropriate notion of a weak solution that requires certain entropy-like conditions. To obtain an approximation of the solution of the nonlinear heat equation, a new conservative first-order finite difference scheme is proposed that respects the corresponding entropy conditions, and certain links between the weak solution and the numerical scheme are provided. Finally, a traffic simulation scenario and a comparison with the Lighthill-Witham-Richards (LWR) model are provided, illustrating the benefits of the use of automated vehicles.