Extended Numerical Fluxes For The Anistropic Traffic Flow Model And Their Applications

The anisotropic high-order traffic flow model which is consistent with the characteristic that disturbances only propagate upstream can describe equilibrium and nonequilibrium traffic flows.It is more reasonable for the description of traffic flow phenomena more than other macroscopic models,and it is a kind of very important macroscopic traffic flow model.However,the anisotropic high-order traffic flow model is essentially nonlinear partial differential equations,and its analytical solution is usually difficult to obtain.For better simulating actual traffic phenomena and exploring the profound mechanism of traffic flow problems,it is necessary to study the theory and algorithm of the anisotropic high-order traffic flow model and seek its accurate and efficient numerical solution.This dissertation focuses on the anisotropic traffic flow model.By exploiting the properties of Riemann invariants in the anisotropic traffic flow model,a number of numerical fluxes are generalized to design the first or high-order numerical schemes.A general equation form is also derived for the modeling of traffic flow under inhomogeneous road conditions,along with the numerical schemes for the solution.Moreover,a new method is proposed to solve the Riemann problem of the anisotropic traffic flow model at a junction,which is crucial for the simulation of traffic flow on a road network.The main contents of the dissertation are briefed as follows.Firstly,the development of the anisotropic traffic flow model is briefly reviewed,as are numerical methods for solving the anisotropic traffic flow model.By studying the propagation properties of Riemann invariants,the classical numerical fluxes such as the Godunov,EO(Engquist-Osher)and LF(Lax-Friedrichs)flux are generalized for the anisotropic high-order traffic flow model;moreover,these generalized fluxes are adopted for the construction of the first-order monotone numerical schemes and the high-order numerical schemes.Numerical experiments indicate that the schemes based on these generalized fluxes are reasonable and efficient.Secondly,by taking into account inhomogeneous road conditions,the discontinuous flow functions are introduced in the anisotropic high-order traffic flow model,and the general form of the anisotropic high-order traffic flow model with discontinuous fluxes is deduced.Based on the properties of the Riemann invariants,the generalized classical numerical fluxes for the anisotropic high-order traffic flow model with discontinuous fluxes are obtained by locally simplifying their homogeneous systems and using the ? mapping algorithm.These numerical fluxes are used to construct the first-order monotone numerical schemes and the high-order WENO(weighted essentially non-oscillatory)schemes.Then these schemes are used to solve the anisotropic high-order traffic flow model with discontinuous fluxes,and expected numerical results are obtained.Lastly,a new method is proposed to solve the Riemann problem of the LWR model at a junction.Usually,this involves an optimization problem which maximizes the sum of outflows on all upstream roads.However,we generalize the optimization problem by maximizing a linear combination of the outflows.The new method is also used to solve the Riemann problem at a junction of the CHO(conserved higher-order)model,thus we reasonably maximize the total outflow at the junction,which is compared with the maximization of total pseudo-outflow in the literature.To sum up,based on transport forms of the model equations and the property which the Riemann invariant propagated by the second characteristic field remains unchanged when crossing the shock or rarefaction,a number of numerical fluxes of the model equations are generalized.Then corresponding first-order and high-order numerical schemes are constructed,which are used to solve traffic flow models on homogeneous roads,inhomogeneous roads,and traffic networks.The research results in this dissertation are of theoretical and practical significance,which are expected to enrich the traffic flow theory and be applied in traffic engineering.