Stability criteria for explicit finite difference solutions of the parabolic diffusion equation with non‐linear boundary conditions

The criteria for stability of the explicit finite difference solution of the one‐dimensional, transient, conduction heat transfer problem with both radiant and convection heat transfer at the boundaries are considered in this paper. These criteria are governed by an inequality set from a functional relationship between the newly calculated and the old temperature at each node. From the node with the most stringent criteria, it is shown that setting the coefficient of the old temperature equal to zero in the governing difference equation is not sufficient for a general criterion. On the other hand, setting the derivative of the new temperature with respect to the old temperature equal to zero in the governing difference equation presents a simple, straightforward technique for obtaining a sufficient condition for a stable system. It is further shown that the second law of thermodynamics, written in explicit finite difference form, does present a necessary criterion for stability. However, the second law, because it is in the form of an inequality, does not present as simple a criterion as the derivative method does. The specific problem studied is a finite thickness slab, initially at a uniform temperature, but instantaneously subjected to both radiation and convection on its two surfaces. Temperature profiles were calculated on a digital computer and are presented in dimensionless graphical form over a range of five dimensionless parameters. A plot that relates stability to the maximum time‐step size for the entire range of practical conditions of radiation numbers is also presented.