Applications of Lax-Friedrich Finite Difference Method Macroscopic and Microscopic Traffic Flow Modelling PDEs and ODEs Numerical Solutions

Abstract

Traffic flow in most urban areas is augmenting due to the growth in transport and continual demand for it. It is multimodal and includes use of different types vehicles, motorcycles andeven walking. The assessment of uninterrupted traffic flow is traditionally based on empirical methods. This study was based on the macroscopic model which is a mathematical model that formulates the relationships among traffic flow characteristics like density, flow, mean and speed of a traffic stream. The study considered traffic models first developed by Lighthill and whitham(1955) and later Richards (1956) shortly Called LW R traffic flow model. Simulation by use of this method enables control strategies of congestion dissipation and has suggested some recommended measures to rationalize the design of roads and implementation of regulations of road users considering some regulations and infrastructural gaps in Kisi town. This paper focus on two finite difference schemes, that is, first order Explicit Upwind Difference Scheme-EUDS (forward time. backward space) and Second order Lax-Wendroff Difference Scheme-LW DS (forward time centred space) for solving first order PDE as well the traffic density ρ(t, x) was computed by solving LW R macroscopic conservation form of traffic flow model using both schemes. The conditions of stability were numerically verified and it is shown that LW DS is superior to EUDS in terms of time step selection. The results obtained were becompared with average key data which provide initial conditions and boundary data used for numerical simulation