An exponentially-fitted method for singularly-perturbed ordinary differential equations with turning points and parabolic problems

An exponentially-fitted method for singularly perturbed, one-dimensional parabolic equations and ordinary differential equations both of the convection-diffusion-reaction type 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-reaction differential operator. This solution is of exponential type and exact for steady, constant-coefficients convection-diffusion-reaction equations. By means of three examples for parabolic equations and two examples for ordinary differential equations with turning points, it is shown that this method provides uniformly convergent solutions with respect to the small perturbation parameter, and these solutions are more accurate and exhibit a higher order of convergence than those obtained with a boundary-layer resolving upwind finite difference scheme on a piecewise-uniform mesh.