On the numerical treatment of an ordinary differential equation arising in one-dimensional non-Fickian diffusion problems

In a recent study, Chen and Liu [Comput. Phys. Comm. 150 (2003) 31] considered a one-dimensional, linear non-Fickian diffusion problem with a potential field, which, upon application of the Laplace transform, resulted in a second-order linear ordinary differential equation which was solved by means of a control-volume finite difference method that employs exponential shape functions. It is first shown that this formulation does not properly account for the spatial dependence of the drift forces and results in oscillatory solutions near the left boundary when these forces are large. A piecewise linearized method that provides piecewise analytical solutions, is exact in exact arithmetic for constant coefficients, homogeneous, second-order linear ordinary differential equations and results in three-point finite difference equations is then proposed. Numerical simulations indicate that the piecewise linearized method is free from unphysical oscillations and more accurate than that of Chen and Liu, especially for large drift forces. The method is then applied to non-Fickian diffusion problems with non-constant drift forces in order to determine the effects of the potential field on the concentration distribution.  2005 Elsevier B.V. All rights reserved. PACS: 02.70.-c; 02.70.Rw; 02.60.Cb; 02.30.Mv