A thesis submitted in partial fulfillment of the requirements for the degree of master?�?s of science in Mathematics

Abstract

This thesis provides a numerical solutions to a scalar advection equation which model to measure the rate of change of the concentration (density) of substances (pollutants) moving through the flow of the fluid in two dimensional space. To solve this equation using finite volume method. Constructing unstructured mesh for irregular shaped cells that needs to calculate the volume(or the area in 2D) for each of different size cells and the time steps to pass each cells make difficult to implement this scheme. Thus to overcome this difficulty the computational domain is embedded in a uniform Cartesian plane which contain most of the computational region in it. And small cells near the border are treated by the same scheme as the interior part. We apply the FV scheme which is modeled by using a wave-propagation approach in which the flux at each cell interface is built up on the basis of information propagating in the upwind direction of this interface from neighboring cells which is unconditionally stable for Courant numbers up to 1. By employing this approach we have shown the stability and convergence of the upwind finite volume scheme to solve linear advection equation through test examples.