A parameter uniform difference scheme for singularly perturbed parabolic problem in one space dimension

A numerical study is made to examine a singularly perturbed parabolic initial-boundary value problem in one space dimension on a rectangular domain. The solution of this problem exhibits the boundary layer on the right side of the domain [H.G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Convection–Diffusion and Flow Problems, Springer-Verlag, New York, 1996]. The Crank–Nicolson finite difference method consisting of an upwind finite difference operator on a fitted piecewise uniform mesh is constructed. The resulting method has been shown almost first order accurate in space and second order in time. We have shown that the resulting method is uniformly convergent with respect to the singular perturbation parameter. Numerical experiments have been carried out, which validate the theoretical results. It is also shown that a numerical method consisting of same finite difference operator on uniform mesh does not converge uniformly with respect to the singular perturbation parameter.