| Traffic flow has been considered to be a continuum flow of a compressible liquid having a certain density profile and an associated velocity, depending upon density, position and time. Several one-equation and two-equation macroscopic continuum flow models have been developed which utilize the fluid dynamics continuity equation and help us find analytical solutions with simplified initial and boundary conditions. In this thesis, the one-equation Lighthill Witham and Richards (LWR) model combined with the Greenshield's model, is used for finding analytical and numerical solutions for four problems: Linear Advection, Red Traffic Light turning into Green, Stationary Shock and Shock Moving towards Right. In all these problems, the numerical solutions are computed using the Godunov Method and the Finite Element Method, and later they are compared to each other. Furthermore, the finite element time relaxation method is introduced for the treatment of the shocks in two numerical problems: (a) Stationary Shock and (b) Shock moving towards the right. The optimal time relaxation parameters are numerically computed using three accuracy measures and finally, the effects of multiple time relaxation settings are explored. |
