| Singular perturbation problems are common in gas and hydrodynamics. Conservation laws of physics lead to Navier-Stokes equations, and convection-diffusion equations are a special form of Navier-Stokes equations at high Reynolds numbers. We develop three high order finite difference techniques for the second order, singularly perturbed linear boundary value problem in one dimension which governs the one dimensional convection-diffusion equation. We use Taylor series expansions and error conversions for the development of the techniques. We state and prove convergence and stability conditions of these techniques, and give some numerical results. We then use one of our techniques on a nonlinear ordinary differential test problem. Finally, we extend the idea of using Taylor series expansions and error conversions to a two dimensional linear test problem.; The most striking point in all the developments in the one dimensional case is the need of using only three points to obtain high accuracy. In the two dimensional case, we use nine points. |
