| Physical and biological processes such as contaminant transport, population dispersal and nerve impulse propagation can be described by diffusion equations with convection/reaction. This dissertation deals with studying such diffusion equations employing various analytical and numerical techniques. In the case of contaminant transport with nonlinear adsorption, we describe the concentration profiles qualitatively using phase plane methods and are able to find exact solutions. We also study the use of the numerical scheme known as the waterbag method, on Fisher-type equations, and compare it to an explicit scheme. When the convective transport equation is in two dimensions, we develop an ADI method with exponential upwinding and carry out extensive numerical computations. It is found that the ADI method with exponential upwinding can be a very useful scheme for convection dominated diffusion problems in more than one dimension. Finally, we also develop a new iterative scheme to solve linear systems of equations, especially when the coefficient matrices are non symmetric and non diagonal dominant. We prove that the convergence region of this new method is better than either Jacobi or Gauss-Seidel iterative method and demonstrate it on some examples where these other methods fail. |
