A Fitted Mesh Method For Partial Differential Equations With A Time Lag

Abstract

A class of initial boundary value problem for singularly perturbed parabolic partial differential equation of convection diffusion type having dominating delayed convection term is examined on a rectangular domain. When the perturbation parameter specifying the problem tends to zero, a breakdown of singular perturbation occurs in narrow intervals of space and short interval of time and the solution of the perturbed problem will often behave analytically quite differently. In these narrow regions which are usually referred as boundary regions, the solution changes rapidly and form parabolic boundary layers in the neighborhood of the outflow boundary regions. Due to the presence of the perturbation parameter and in particular time delay classical numerical method on uniform meshes are known to be inadequate for the solution of such type of problems, because the error in the numerical solution depends appreciably on the value of the perturbation parameter and is comparable with the solution itself for small value of perturbation parameter. Thus in connection with the stiff behavior it is of interest to develop special numerical methods whose errors would be independent of the perturbation parameter, i.e. parameter-uniformly convergent methods. In this paper, a numerical method consisting of standard upwind finite difference operator on a piecewise uniform mesh is constructed, in order to overcome the difficulties due to the presence of perturbation parameter and the time delay. The first step in this direction consists of discretizing the time variable with the backward Euler's method with constant time step. This produces a set of stationary singularly perturbed semidiscrete problem class which is further discretized in space using upwind finite difference operator on a piecewise uniform mesh. An extensive amount of analysis is carried out in order to establish the convergence and stability of the method proposed. The analysis in this paper uses a suitable decomposition of the error into smooth and singular component and a comparison principle combined with appropriate barrier functions. The error estimates are obtained which proves uniform convergence of the method.