Adaptive Numerical Method for Approximation of Traffic Flow Equations

Abstract

For a long time now, traffic equations have been considered, and different modeling has been done for it. In this article, we work on the macroscopic model, especially the most famous light model. Because these models are among the stiff and shocking problems, theoretical methods do not give good answers to these problems. This paper describes a meshless method to solve the traffic flow equation as a stiff equation. In the proposed method, we also use the exponential time differencing (ETD) method and the exponential time differencing fourth-order Runge–Kutta (ETDRK4). The purpose of this new method is to use methods of the moving least squares (MLS) method and a modified exponential time differencing fourth-order Runge–Kutta scheme. To solve these equations, we use the meshless method MLS to approximate the spatial derivatives and then use method ETDRK4 to obtain approximate solutions. In order to improve the possible instabilities of method ETDRK4, approaches have been stated. The MLS method provided good results for these equations due to its high flexibility and high accuracy and has a moving window and obtains the solution at the shock point without any false oscillations. The technique is described in detail, and a number of computational examples are presented.