Study On Covolume-Upwind Finite Volume Approximations For Linear Parabolic Partial Differential Equations

In this thesis we solve two-dimensional linear parabolic partial differential equations with pure Dirichelet boundary conditions, using the bilinear covolume-upwind finite volume method on rectangular grids to discretize the spatial variables and the Crank-Nicholson method for the time variable. These PDEs provide a model for problems from various fields of engineering and applied sciences, such as unsteady viscous flow problems, the simulation of oil extraction from underground reservoirs, transport of air and ground water pollutants and modeling of semiconductor devices. Finite volume method has the important advantage of allowing the conversion of integrations over the control volume to integrations over its boundary based on Green\u27s Theorem. Then, one can use quadrature rules to approximate the resulting integrals. In order to avoid non-physical oscillations that can arise from the numerical solution of convection-dominated problems when using the central finite volume scheme, we generate non-standard control volumes using local Peclet\u27s numbers and the upwind principle. We numerically compare the covolume-upwind finite volume method with the central and the upwind finite volume schemes, demonstrating stability and better convergence of the method through various examples