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Numerical Method for Solving the Diffusion Equation: I. Horizontal Flow in Semi-Infinite Media¹

ABSTRACT

The diffusion equation has been successfully used by several investigators to describe moisture movement in soil. Their success depends largely upon the Boltzmann transformation which can only be used for semi-infinite, uniform regions.

A numerical technique is developed for solving the implicit difference analogues of the diffusion equation. This method involves (1) devising suitable difference analogues for the differential equation, (2) developing a method for choosing diffusivity to convert the nonlinear difference analogues into linear equations, and (3) solving the system of linear equations.

The numerical method can be applied to finite, nonuniform media. However, the study reported is limited to one-dimensional, horizontal flow in a uniform, semi-infinite medium so that the solutions obtained can be compared to solutions obtained by the Boltzmann transform technique. The results of the study indicate that the numerical methods and the Boltzmann transform techniques give very similar solutions and that solutions obtained by both methods are similar to experimental results.

¹ Technical Paper No. 1467 Oregon Agr. Exp. Sta., Corvallis. Research contribution to Regional Project W-68.

² Formerly Junior Soil Scientist, Oregon State University, now Assistant Professor, Utah State University, Logan; and Research Assistant, Associate Professor, and Assistant Professor, Oregon State University, Corvallis, respectively.

Received for publication October 23, 1961. Accepted for publication June 21, 1962.