An exponentially-fitted method for singularly perturbed, one-dimensional, parabolic problems

An exponentially-fitted method for singularly perturbed, one-dimensional, linear, convection-diffusion-reaction equations in equally-spaced grids is presented. The method is based on the implicit discretization of the time derivative, freezing of the coefficients of the resulting ordinary differential equations at each time step, and the analytical solution of the resulting convection-diffusion differential operator. This solution is of exponential type and exact for steady, constant-coefficients convection-diffusion equations with constant sources. By means of three examples, it is shown that this method provides uniformly convergent solutions with respect to both the time step and the small perturbation parameter, and these solutions are more accurate and exhibit a higher order of convergence than those obtained with an upwind finite difference scheme in a piecewise uniform mesh that is boundary-layer resolving.