Uniformly convergent numerical method for singularly perturbed differential-difference equation using grid equidistribution

In this paper, a class of singularly perturbed differential-difference equations with small delay and shift terms is considered. A numerical method comprising of upwind finite difference operator on an adaptive grid, which is formed by equidistributing the arc-length monitor function, is constructed for approximating the solution. The method is proved to be robust, in the sense that the discrete solution obtained converges in the maximum norm to the exact solution uniformly with respect to the perturbation parameter. Parameter-uniform error bounds for the numerical approximations are established. Numerical examples support the theoretical results. Copyright (C) 2010 John Wiley & Sons, Ltd.