Comparison of Numeric to Analytic Solutions for a Class of Nonlinear Diffusion Equations

Abstract

In order to test the accuracy of numerical algorithms for solving nonlinear diffusion equations, self-similar analytic solutions to a class of such equations have been constructed using Lie group techniques. The diffusion equations treated have the form δtk = 'δx(g'knδxk), where δt and δx represent partial derivatives with respect to time and position, k is the diffusing quantity, n is the nonlinearity parameter, and g is a general scaling constant. A standard time-implicit finite-difference diffusion algorithm is shown to give excellent agreement with the analytic solutions if the time and position step sizes are sufficiently small. In particular, an automatic time-step control which limits the fractional change ink produces quite good results. Examples are presented using several fixed time and position step sizes for four values of the nonlinearity parameter (n = 1/2,1, 3/2, 2) and using the automatic time-step control for three values (n = 1/2, 1, 3/2).