Numerical modeling of a first-order traffic flow approach with the finite difference-based lines method

Traffic congestion leads to environmental and societal problems. Traffic flow theory is a part of transportation that is concerned with the capacity of transportation facilities and the quality of traffic operation. The goal of traffic flow theory is to evaluate a traffic flow's operational quality under a set of reasonable criteria. Mathematical models for traffic flow can be classified into three main categories: macroscopic, mesoscopic and microscopic approaches. Macroscopic traffic flow models have been created by analogy with fluid dynamics and the resulting partial differential equations are effective in modelling complex traffic events. The interactions between traffic flow parameters including density, flow, and speed are formulated by macroscopic traffic flow models. In this study, the numerical solution of a macroscopic first-order hydrodynamic traffic flow model is carried out using the finite difference-based method of lines approach. With the numerical method, firstly, the spatial variable of the partial differential equation is discretized with finite difference approximations of having orders of p = 1,2,…,8 to obtain a semi-discretized ordinary differential equation system. Four different methods are adaptively used for the temporal discretization of the resultant differential equations: explicit Euler, explicit midpoint, linear implicit Euler, and linear implicit modified midpoint. The performance of the finite difference-based method of lines approach is first assessed by solving the Burgers equation for which an analytical solution exists. The numerical solution yield highly accurate and stable solutions for this problem. The results obtained also played an instructive role in determining the finite difference parameters for computing the required reference solution of the traffic flow model. Within detailed numerical experiments, the accuracy, h-stability, p-stability, and computation time of the finite difference-based method of lines approach for the first-order traffic flow model were investigated. Both implicit and explicit techniques yielded highly accurate results with the appropriate selection of the maximum time step value. The results obtained show that implicit temporal discretization methods have h-stability for all p values, while the h-stability of explicit-time discretization methods is related to the selection of the maximum time step. p-stability for implicit methods is obtained when the number of spatial discretization points is more than 100. A comparison between the CPU times of explicit and implicit methods reveals that the explicit methods are advantageous to implicit approaches. It is possible to decrease the computation time of the method significantly by decreasing the tolerance values used for adjusting the adaptive temporal steps. Within the scope of the first-order traffic flow model, the time-dependent partial differential equation is solved efficiently with the numerical method under different boundaries and initial conditions. In addition, simulations were performed with different values of factors that determine driver behavior and road quality in the first order model, and thus, a traffic flow modeling that takes psycho-mechanistic effects into account is successfully carried out with the finite difference-based method of lines approach.