An Implicit Finite Difference Scheme with Preconditioning for Convection Dominated Diffusion Equation

An implicit finite difference scheme for solving time-dependent convection dominated diffusion equations in two space variables is presented. A one-sided difference approximation is used for the convection terms and a second-order central difference approximationfor the diffusion term. The implicit scheme is consistent,unconditionally stable and its $L^2$ error estimation is optimal. By preconditioning technique of incremental unknowns, the implicit scheme is an efficient one.