The impact of source terms in the variational representation of traffic flow

TL;DR

This paper analyzes how source terms affect the variational theory of traffic flow, revealing existence conditions and proposing improved numerical methods for cases with exogenous sources.

Contribution

It clarifies the conditions for variational theory solution existence with source terms and introduces enhanced numerical methods for exogenous sources.

Findings

VT solutions exist only with exogenous source terms in Eulerian coordinates

Discrete time makes source dependencies effectively exogenous

VT solutions may not exist in Lagrangian coordinates with source terms

Abstract

This paper revisits the variational theory of traffic flow, now under the presence of continuum lateral inflows and outflows to the freeway say Eulerian source terms. It is found that a VT solution exists only in Eulerian coordinates when source terms are exogenous meaning that they only depend on time and space, but not when they are a function of traffic conditions, as per a merge model. In discrete time, however, these dependencies become exogenous, which allowed us to propose improved numerical solution methods. In Lagrangian and vehicle number-space coordinates, VT solutions may not exist even if source terms are exogenous.