  We propose a bound-preserving Runge-Kutta (RK) discontinuous Galerkin (DG)
method as an efficient, effective and compact numerical approach for numerical
simulation of traffic flow problems on networks, with arbitrary high order
accuracy. Road networks are modeled by graphs, composed of a finite number of
roads that meet at junctions. On each road, a scalar conservation law describes
the dynamics, while coupling conditions are specified at junctions to define
flow separation or convergence at the points where roads meet. We incorporate
such coupling conditions in the RK DG framework, and apply an arbitrary high
order bound preserving limiter to the RK DG method to preserve the physical
bounds on the network solutions (car density). We showcase the proposed
algorithm on several benchmark test cases from the literature, as well as
several new challenging examples with rich solution structures. Modeling and
simulation of Cauchy problems for traffic flows on networks is notorious for
lack of uniqueness or (Lipschitz) continuous dependence. The discontinuous
Galerkin method proposed here deals elegantly with these problems, and is
perhaps the only realistic and efficient high-order method for network
problems.
