A piecewise-analytical method for singularly perturbed parabolic problems

A piecewise-analytical 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 reaction-convection-diffusion differential operator. This solution is of exponential type, accounts for the convection, diffusion, reaction and transient effects, and is 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 the small perturbation parameter, but, if the time step is sufficiently large, then the roots of the characteristic polynomial that defines the exponential solutions of the homogeneous differential equation may become nearly independent of the time step and the solution may exhibit large errors. It is also shown that a rational use of piecewise-analytical methods for small values of the perturbation parameter requires a knowledge of the characteristic times that control the evolution of the parabolic problem that is to be solved.