Runge-Kutta Discontinuous Galerkin Method for Traffic Flow Model on Networks

摘要

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