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

Abstract

Abstract. 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.

Discontinuous Galerkin Method for Material Flow Problems

“…However, due to the underlying geometry of the problem discontinuous Galerkin methods represent an alternative approach. This has been successfully shown in [17][18][19][20][21][22][23] for similar structured problems in one dimension. Our intention is now to derive a discontinuous Galerkin method in two space dimensions to solve the nonlocal hyperbolic conservation law from [5].…”

Modeling vehicular traffic networks. Part I

“…Although the matrix M determines the stationary matrix C 1 , the inverse is not possible, the knowledge of C 1 does not permit the full determination of M. This is the essence of the irreversible evolution. Writing one row of matrix (26) as the vector Z T = m 1 m 2 · · · m N −1 m N , it can be noted that it is a stationary distribution because Z T M = Z T (eigenvalues 1), or C 1 M = C 1 . From Example 1, Eqs.…”

Robust Traffic Density Estimation Using Discontinuous Galerkin Formulation of a Macroscopic Model

“…In an effort to keep the following material as compact as possible, we keep the detail to a bare minimum. For further details on Spectral methods, see [15] and [16], and for previous work on the DG method applied to a traffic flow model, see [17] and [18].…”

“…However, our choice of numerical flux, (17), would make this difficult to do. We also note that each of the elemental systems should be solved separately for the sake of computational efficiency.…”