On diffusive methods and exponentially fitted techniques

Liouville's transformations are employed to reduce a one-dimensional, time-dependent convection-diffusion-reaction operator to a diffusion-reaction one which upon a control-volume, second-order accurate discretization results in the same finite difference methods as those of exponentially fitted techniques. Two variants of these techniques are considered depending on the continuity and/or smoothness of the analytical albeit approximate solution to the ordinary differential equations that result upon discretization of the time variable, time linearization, and piecewise spatial linearization of linear equations. It is shown that these exponential techniques include those of Allen and Southwell, Il'in, and El-Mistikawy and Werle for steady, linear convection-diffusion operators, and are linearly stable and monotonic. It is also shown that exponentially fitted techniques account for the characteristic times of reaction, diffusion and residence, and the time step employed in the discretization of the time variable, can use adaptive refinement techniques, and can account for the convection, diffusion and reaction, the convection and diffusion, the reaction and diffusion, and the diffusion processes. However, the latter three require iterative techniques to find the numerical solution.