Elsevier

Applied Mathematics and Computation

Volume 244, 1 October 2014, Pages 567-576
Applied Mathematics and Computation

Discontinuous Galerkin finite element scheme for a conserved higher-order traffic flow model by exploring Riemann solvers

https://doi.org/10.1016/j.amc.2014.07.002Get rights and content

Abstract

The discontinuous Galerkin (DG) scheme is used to solve a conserved higher-order (CHO) traffic flow model by exploring several Riemann solvers. The second-order accurate DG scheme is found to be adequate in that the accuracy is comparable to the weighted essentially non-oscillatory (WENO) scheme with fifth-order accuracy and much better than the scheme with first-order accuracy in resolving a wide moving jam with a shock profile. Moreover, it considerably reduces the differences between the proposed solvers in generating numerical viscosities or errors. Thus, this scheme can maintain high efficiency when a simple solver is adopted. The scheme could be extended to solve more complex problems, such as those related to traffic flow in a network.

Introduction

A Riemann solver, or numerical flux, is sufficient for designing a first-order accurate scheme for hyperbolic conservation laws. The Godunov, Engquist-Osher (EO), and Lax-Friedrichs (LF) are well-known numerical fluxes for solving scalar equations, such as the Lighthill-Whitham-Richards (LWR) model [16], [21] and the Burgers equation, which give rise, respectively, to the Godunov, EO, and LF schemes [4], [6], [23], [24] (see also [2], [3], [5], [8], [9], [11], [15], [26], [29], [35] for studies of the LWR or Multi-class LWR models). To solve the conserved higher-order (CHO) model [33], which is an extension of the LWR model, the aforementioned fluxes were extended with a so-called traffic flow (TF) flux to design a number of first-order accurate schemes [30]. Although these extended fluxes were able to reproduce stable and convergent numerical solutions, considerable numerical viscosities or errors were observed.

This sequence simplifies the expression of these fluxes and the computation becomes more efficient; however, the shock profile of the reproduced wide moving jam in the CHO model becomes smoother and the backward moving wave becomes faster, which suggests an increase in numerical viscosity. It seems that the improvement in efficiency from computing a simpler numerical flux is counteracted by an increase in numerical viscosity. If a first-order accurate scheme is adopted, the grid needs to be refined to improve the solution. Generally, improving the scheme’s accuracy should suppress the redundant numerical viscosity in a simpler flux. More precisely, numerical fluxes (either simple or complicated) are expected to produce similarly accurate solutions in a higher-order scheme. In this case, using a simpler flux will improve the scheme’s efficiency.

Following this insight, this paper designs a discontinuous Galerkin (DG) scheme for solving the CHO model by exploiting the numerical fluxes extended from the Godunov, EO, LF and TF fluxes [19], [30]. The DG scheme is found to be more accurate than the first-order scheme in that it is able to suppress differences between the explored solvers when generating numerical viscosities. Moreover, we find that the performance of the second-order DG scheme is similar to that of the fifth-order WENO schemes, a similarity that is due to the existence of a shock profile in the solution, although the latter scheme is expected to be more accurate in generating smooth solutions. These findings indicate that the second-order DG scheme is adequate for the model, especially when a simpler flux with a lower computational cost is adopted. To enhance the argument, an analytical solution for a wide moving jam is constructed for comparison with the numerical solutions.

The remainder of this paper is organized as follows. Section 2 discusses the CHO model, together with its solution for a wide moving jam. In Section 3, the extended Godunov, EO, LF and TF numerical fluxes are derived by applying the Riemann solution to the model’s homogeneous system. In Section 4, the DG schemes are constructed using these fluxes. In Section 5, numerical solutions to the wide moving jam are derived through simulation, and the solutions produced by the first-order accurate scheme, the fifth-order accurate WENO scheme and the second-order accurate DG scheme are compared with the analytical solution. Section 6 concludes the paper.

Section snippets

Model equations

Taken as a continuum, the mass conservation of traffic flow is described through the following partial differential equation:tρ+x(ρv)=0,where ρ(x,t) and v(x,t) are the density and velocity in location x at time t, respectively, and q(x,t)=ρ(x,t)v(x,t) denotes the flow at (x,t). Assume that there is a determined equilibrium velocity-density relationship v=ve(ρ) (ve(ρ)<0). Then, the substitution of v in Eq. (1) by this equation gives rise to the following well-known LWR model [16], [21]:tρ+x(

Riemann solvers for the homogeneous system

System (3), (4) is rewritten in the following vector form:ut+f(u)x=s(u),with u=(ρ,w)T,f(u)=(f1(u),f2(u))T,f1(u)=ρV(w),f2(u)=wV(w) and s(u)=(0,β-1(V(w)-ve(ρ)))T. To derive a numerical flux, the Riemann problem in the homogeneous systemut+f(u)x=0is of interest. The problem is actually set with the initial valuesu(x,0)=u1,x<0,u2,x>0,where u(x,0)=(ρ(x,0),w(x,0))T,u1=(ρ1,w1)T and u2=(ρ2,w2)T.

The discontinuous Galerkin finite element scheme

By dividing the computational interval (0,L) into cells, Ii=(xi-1/2,xi+1/2), with Δi=xi+1/2-xi-1/2,i=1,,N, we multiply system (12) by a test function w(x) and integrate the resultant equations over Ii. By applying the integration of the parts to the second term, we haveIiu(x,t)tw(x)dx-Iif(u)w(x)dx+f(u(xi+12,t))w(xi+12)-f(u(xi-12,t))w(xi-12)=Iis(u)w(x)dx.

Numerical simulation

In system (3), (4), we generally set the relaxation time as τ=30s, and the velocity-density relationships asV(ρ)=vf1-ρ/ρjam1+b(ρ/ρjam)+a(ρ/ρjam)2,ve(ρ)=vf1+expρ/ρjam-0.250.06-1-3.72×10-6,where vf=25m/s,a=4,b=-0.8, and ρjam=0.16 veh./m. Similar settings were chosen in [19], [30]. To ensure the numerical stability of a scheme, the time step should satisfy the following CFL condition:Δt(n)=CΔxα(n),where α(n)=maxi{|λ1(ρi(n),wi(n))|,|λ2(ρi(n),wi(n))|}.

The computational interval is set as [0,L], with L

Conclusions

The CHO model is composed of the mass and pseudo-mass conservations, which are favorably consistent with the LWR model. Underlying this consistency is the finding that the Riemann solver for the model can be expressed by the classical Godunov flux, which is based on a Riemann invariant that remains unchanged through propagation from the initial state on the upstream side to the interface. Accordingly, the classical EO and LF fluxes, together with a TF flux, are extended, allowing us to obtain a

Acknowledgements

This study was jointly supported by grants from the National Natural Science Foundation of China (11072141, 11272199), the National Basic Research Program of China (2012CB725404), the Shanghai Program for Innovative Research Team in Universities, and a National Research Foundation of Korea grant funded by the Korean government (MSIP) (NRF-2010-0029446).

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