The comparative performance of some finite difference equations for transient heat conduction

Abstract

AbstractThe established finite difference equations to estimate one‐dimensional transient heat flow in solids are the ‘classical’ form (explicit), and the Crank–Nicolson and ‘pure‐implicit’ forms (both implicit). They are all based on finite difference approximation to the Fourier continuity equation. To these are now added three more explicit forms: exponential linear, exponential inverse cosine and polynomial, which are based on exact solutions to the Fourier equation. The performance of each of the six equations is tested against the exact results of a well known step excitations problem (the Groeber model). The tests consist of examining (i) finite difference errors arising from a single implementation of each form at different stages in the transient cooling process, (ii) the errors that accumulate during part or all of the cooling process (both as regards any bias that is introduced, and also a measure of variance) and (iii) the run times in executing the various forms. The nondimensional time step r was treated as the independent variable, and can be made arbitrarily large, by use of a simple time‐division procedure (otherwise r < ½ for use with the classical form). It is shown that having regard to both error and run time, the polynomial form appears to be the most efficient estimator.