Riemann solvers of a conserved high-order traffic flow model with discontinuous fluxes
Introduction
There are some roads where the conditions (e.g., the lane number and the free-flow velocity) vary with the spatial location , which are known as inhomogeneous roads. Traffic flow models on inhomogeneous roads are essentially the hyperbolic balance laws with discontinuous fluxes, which can be written in the following form:where is an unknown scalar or vector for solution, the flux depends on the , which is a known scalar or vector denoting some spatially varying parameters, and is a source term. If , then Eq. (1) is known as the hyperbolic conservation laws with discontinuous fluxes.
There are considerable applications involving spatially varying fluxes (or discontinuous fluxes), e.g., in flow through porous media, elastic waves in heterogeneous media, and traffic flow with inhomogeneous road conditions. The problems with discontinuous fluxes have been considered and developed over recent decades, and many research results have emerged [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]. However, different from the standard hyperbolic balance laws, classical first-order monotone schemes, nonlinear weighted essentially non-oscillatory (WENO) and Runge-Kutta discontinuous Galerkin (RKDG) high-order schemes are difficult to be directly applied in solving problems with discontinuous fluxes [7], [9], [10], [12], [13].
Bale and Leveque [2] developed wave propagation algorithms for solving conservation laws and balance laws with spatially varying flux functions. Zhang and Liu [7], [9] discussed the scalar hyperbolic conservation laws with discontinuous flux and its Riemann solver. In addition, a -mapping algorithm,in which is “frozen” on the interface (grid boundary), was proposed, so that is locally independent of the parameter (or ). Therefore, a classical numerical flux, e.g., the Godunov, Lax-fridrichs (LF) or Engquist-Osher (EO) flux can be used to design the corresponding first-order scheme. Moreover, the -mapping algorithm can be respectively combined with WENO and RKDG methods to construct high-order WENO and RKDG schemes, see Refs. [10], [13], [16], [24]. Zhang et al. [12] regarded as a solution variable, converted the multi-class LWR model with inhomogeneous road conditions to the hyperbolic conservation laws, and obtained the numerical solution by the WENO scheme. Bürger et al. [14], [15], [19] intensively studied a class of multi-species kinematic flow models with discontinuous fluxes, and designed a number of difference schemes. Jin et al. [18] introduced supply-demand diagrams and presented a new framework for analyzing the inhomogeneous LWR model. Wang [20] proposed an extended EO monotone difference scheme for the scalar hyperbolic conservation laws with discontinuous flux. Chen et al. [21] put forward a high-resolution relaxation scheme to solve the multi-class LWR model with inhomogeneous road conditions. Wines et al. [22] used a mollifier to smooth out the discontinuity in the flux function, and obtained the Riemann solver for a kinematic wave traffic model with discontinuous flux. Wang and Hu [25] discussed an interface Roe-type numerical flux function and a new finite difference scheme for the scalar conservation laws with discontinuous flux. Chabot et al. [26] proposed a nodal high-order discontinuous Galerkin method for 1D wave propagation in nonlinear heterogeneous media. Sun et al. [28] proposed a new modified Local Lax-Friedrichs scheme for the scalar conservation laws with discontinuous flux.
As mentioned above, there are numerous studies of traffic flow models with discontinuous fluxes. However, these works mainly considered the LWR model with discontinuous flux [7], [8], [9], [10], [11], [14], [17], [18] and the multi-class LWR model with discontinuous fluxes [12], [15], [16], [19], [21], [24]. For the higher-order traffic flow model with discontinuous flux, Li [3] studied the well-posedness theory of an isotropic higher-order traffic flow model with inhomogeneous road conditions. Zhang et al. [11] extended the isotropic Zhang model [29] to a high-order traffic flow model with variable lanes and free flow velocities.
Generally speaking, the LWR model with discontinuous flux and the multi-class LWR model with discontinuous fluxes can only describe the equilibrium traffic flow. The high-order traffic flow model, which is classified into isotropic and anisotropic models, can describe equilibrium as well as non-equilibrium traffic flows. In isotropic models, the speed of traffic flow is between the two characteristic speeds, which could result in unreasonable vehicle reverse flows [30]. Thus, the anisotropic high-order models have been widely studied in recent years. Zhang et al. [31] proposed an anisotropic high-order model, which has a conservative system of equations. Therefore, the model is named as a conserved higher-order anisotropic traffic flow model. Due to the lack of studies on anisotropic high-order traffic flow models with inhomogeneous road conditions, this paper develops a conserved high-order anisotropic traffic flow model with inhomogeneous road conditions, and discusses its Riemann solvers.
The remainder of this paper is organized as follows. Section 2 discusses the CHO model with discontinuous fluxes, together with its hyperbolicity, characteristic fields and Riemann solvers to the homogeneous system. In Section 3, the first-order Godunov scheme is designed to simulate wave breaking patterns and bottleneck effects in traffic flow. Moreover, we discuss the invariant region principle of numerical solutions. Section 4 concludes the paper.
Section snippets
Model equations
In the LWR theory, the traffic flow is considered as a compressible fluid [32], [33]. Therefore, the mass conservation of traffic flow is described through the following partial differential equation:where and are the density and velocity in location at time , respectively, and denotes the flow at . Let and be the number of lanes and the free flow velocity in location . Then, is the density in a single lane, which is
Numerical scheme
The interval is uniformly divided into cells: , with , , . For a division of the time interval , where and , the first-order Euler forward time discretization is used. Therefore, the first-order Godunov scheme for the system of (3) and (4) is as follows:where , ,
Conclusion
Considering inhomogeneous road conditions, we extend the CHO model to the CHO model with discontinuous fluxes, and discuss its hyperbolicity and characteristic fields in detail. According to the analysis of the Riemann problem for its homogeneous subsystem and the property of its Riemann invariants, we illustrate wave breaking patterns and obtain the exact Riemann solvers to the homogeneous system of this model. Moreover, we give the first-order Godunov scheme and discuss the invariant region
Acknowledgments
This study was jointly supported by grants from the National Key R&D Program of China under Grant no. 2018YFB1600900, the National Natural Science Foundation of China under Grant nos.72021002, 71890973, 11972121, and 62003182.
References (49)
- et al.
Hyperbolic conservation laws with space-dependent flux: I. Characteristics theory and Riemann problem
J. Comput. Appl. Math.
(2003) - et al.
Hyperbolic conservation laws with space-dependent fluxes: II. General study of numerical fluxes
J. Comput. Appl. Math.
(2005) - et al.
A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway
J. Comput. Phys.
(2006) - et al.
-mapping algorithm coupled with WENO reconstruction for nonlinear elasticity in heterogeneous media
Appl. Numer. Math.
(2007) - et al.
A hybrid scheme for solving a multi-class traffic flow model with complex wave breaking
Comput. Methods Appl. Mech. Eng.
(2008) An Engquist–Osher type finite difference scheme with a discontinuous flux function in space
J. Comput. Appl. Math.
(2011)- et al.
Riemann solver for a kinematic wave traffic model with discontinuous flux
J. Comput. Phys.
(2013) - et al.
A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations
Appl. Math. Comput.
(2016) - et al.
A Runge–Kutta discontinuous Galerkin scheme for hyperbolic conservation laws with discontinuous fluxes
Appl. Math. Comput.
(2017) - et al.
The roe-type interface flux for conservation laws with discontinuous flux function
Appl. Math. Lett.
(2018)
A high-order discontinuous Galerkin method for 1D wave propagation in a nonlinear heterogeneous medium
J. Comput. Phys.
High order well-balanced finite difference WENO schemes for shallow water flows along channels with irregular geometry
Appl. Math. Comput.
A new modified local Lax–Friedrichs scheme for scalar conservation laws with discontinuous flux
Appl. Math. Lett.
A theory of nonequilibrium traffic flow
Transp. Res. B
Requiem for second-order fluid approximations of traffic flow
Transp. Res. B
A conserved higher-order anisotropic traffic flow model: Description of equilibrium and non-equilibrium flows
Transp. Res. B
The Aw–Rascle and Zhang’s model: Vacuum problems, existence and regularity of the solutions of the Riemann problem
Transp. Res. B
Well-posedness for a class of conservation laws with data
J. Differ. Equ.
Stability of conservation laws with discontinuous coefficients
J. Differ. Equ.
Discontinuous Galerkin finite element scheme for a conserved higher-order traffic flow model by exploring Riemann solvers
Appl. Math. Comput.
Steady-state traffic flow on a ring road with up- and down-slopes
Phys. A
The Godunov scheme and what it means for first order traffic flow models
A wave propagation method for conservation laws and balance laws with spatially varying flux functions
SIAM J. Sci. Comput.
Well-posedness theory of an inhomogeneous traffic flow model
Discrete Contin. Dyn. B
Cited by (4)
A triangle-based positive semi-discrete Lagrangian–Eulerian scheme via the weak asymptotic method for scalar equations and systems of hyperbolic conservation laws
2024, Journal of Computational and Applied MathematicsA geometrically intrinsic lagrangian-Eulerian scheme for 2D shallow water equations with variable topography and discontinuous data
2023, Applied Mathematics and ComputationCitation Excerpt :The presence of non-autonomous flux functions yields discontinuous local Riemann problems and complicate the identification of the correct wave structure and interaction that is needed in Godunov-type Finite Volume (FV) or Discontinuous Galerkin (DG) discretization approaches (see, e.g., [26–30]). Most of the approaches used in the presence of discontinuous fluxes are tailored to the specific case of study and difficulties have been reported to adapt these approaches to general first order monotone schemes [30]. On the other hand, the presence of flux variables in the source term also requires careful treatment, in particular to ensure the well-balance of the discrete equations [14,31].
Higher-Order Traffic Flow Model Extended to Road Networks
2023, Journal of Transportation Engineering Part A: Systems