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

•Variational solutions are no valid in general; only in Eulerian coordinates when source terms are exogenous.•In discrete time, however, these dependencies become exogenous, which allowed us to propose improved numerical solution methods.•In space-Lagrangian and time-Lagrangian variational solutions do not exist even if source terms are exogenous.

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