An exponentially fitted spline method for singularly perturbed parabolic convection-diffusion problems with large time delay

Abstract

This paper deals with the numerical solutions of singularly perturbed parabolic convection-diffusion problems with a large delay in time. An exponentially fitted scheme is formulated using cubic splines method in equally-spaced grids. As the delay parameter is larger than the perturbation parameter, a special type of mesh is used for the temporal variable so that the shift lies on the nodal points and an exponentially fitted cubic spline method based on Crank-Nicolson's discretization scheme is developed. The obtained scheme is conditionally stable. Parameter-uniform error estimates are derived and it is shown that the method is ε-uniformly convergent of second-order accurate in both temporal direction and spatial direction. The proposed method provides more accurate solutions than an upwind finite difference scheme in a piecewise uniform mesh.