Practical Techniques for Estimating the Accuracy of Finite-Difference Solutions to Parabolic Equations

Abstract

A criterion is proposed which provides a priori means of choosing increments in the independent variables for effecting accurate finite-difference solutions to parabolic partial-differential equations. This is accomplished by relating the increments to the thickness of the diffusion layer. In this manner, the size of the increments is related to the magnitude of the derivatives which are known to influence strongly the accuracy. The fac- that the thickness of the diffusion layer is unknown is surmounted by relating the parameters of the discrete, physical plane to the diffusion variable and to the diffusion thickness. The diffusion variable is a dimensionless coordinate which governs the diffusion process. In similar problems, the diffusion variable is identical to the independent similarity variable. The thickness of the diffusion layer in terms of the diffusion coordinate is shown to be of the same order of magnitude for a wide variety of problems. The utility of the proposed criterion is demonstrated with numerous finite-difference solutions to problems in the areas of heat conduction and boundary-layer theory.