Eulerian-Lagrangian localized adjoint methods (ELLAM) are a class of characteristic methods that have been shown to be effective in solving advection-dominated transport equations. In this thesis, we present an ELLAM method for the solution of the two-dimensional transient advection diffusion equation which uses Wachspress type rational basis functions over non-uniform pentagonal discretizations of the spatial domain. The derived scheme maintains the advantages of previous ELLAM methods and generates accurate numerical solutions for the model equations even when large time steps are used in the simulation. Numerical experiments are presented to illustrate the performance of this method and to investigate its convergence numerically using standard test problems with known analytical solutions.