  While macroscopic models for single or multi-lane traffic flow are well
established, these models are not applicable to the dynamics and
characteristics of disordered traffic which is characterized by widely
different types of vehicles and no lane discipline. We propose a first-order
two-dimensional Lighthill-Whitham-Richards (LWR) model for the continuous
macroscopic longitudinal and lateral dynamics of this type of traffic flow. The
continuity equation is extended into two dimensions and the equation is closed
by assuming a longitudinal flow-density relationship as in traditional
one-dimensional models while the lateral dynamics is based on boundary
repulsion and a desire of a majority of the drivers to go to less dense
regions. This is equivalent to Fick's law giving rise to a lateral diffusion
term. Using the proposed model, several numerical tests were conducted under
different traffic scenarios representing a wide range of traffic conditions.
Even for extreme initial conditions, the model's outcome turned out to be
plausible and consistent with observed traffic flow dynamics. Moreover, the
numerical convergence test is performed using an analytical solution for
lateral steady-state conditions. The model was applied for bicycle simulation
and reproduced the evolution of lateral density profile with asymmetric
behavior.
