  This article proposes a numerical scheme for computing the evolution of
vehicular traffic on a road network over a finite time horizon. The traffic
dynamics on each link is modeled by the Hamilton-Jacobi (HJ) partial
differential equation (PDE), which is an equivalent form of the
Lighthill-Whitham-Richards PDE. The main contribution of this article is the
construction of a single convex optimization program which computes the traffic
flow at a junction over a finite time horizon and decouples the PDEs on
connecting links. Compared to discretization schemes which require the
computation of all traffic states on a time-space grid, the proposed convex
optimization approach computes the boundary flows at the junction using only
the initial condition on links and the boundary conditions of the network. The
computed boundary flows at the junction specify the boundary condition for the
HJ PDE on connecting links, which then can be separately solved using an
existing semi-explicit scheme for single link HJ PDE. As demonstrated in a
numerical example of ramp metering control, the proposed convex optimization
approach also provides a natural framework for optimal traffic control
applications.
