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1. Introduction
The application of the transmission line matrix method (TLM) to electromagnetic problems is well established. 1-5 More recently TLM has been applied to problems in diffusion.6 TLM has been applied successfully to thermal diffusion in one, and three dimensions.8 Results have been found to be in good agreement with those derived using more traditional techniques. It is not our intention to describe the operation of a TLM diffusion model here, since this has been done in detail e l s e ~h e r e . ~J ~)
The relationship of TLM to other techniques has been established." Briefly, traditional techniques fall into one of two categories-those which are explicit but not unconditionally stable, and those which are unconditionally stable but implicit. Among the traditional techniques, the only exception to this classification is the Du Fort-Frankel method, which is explicit and unconditionally stable. Unfortunately, the Du Fort-Frankel method is two-step. This means that, calculations at iteration n require information from iteration (n-1) and iteration (n-2). Retaining values from iteration (n-2) necessitates additional storage relative to a one-step technique, where iteration n requires information from iteration (n-1) only. TLM is explicit, onestep and unconditionally stable. This unique combination of properties means that a numerical routine based on TLM generally requires only a small fraction of the computing resources needed by, for example, its finite element or finite difference counterpart.
In TLM, if the iteration timestep is increased, accuracy decreases but numerical stability is not lost. Throughout the course of most modelling situations, as the rate of temperature change alters, the timestep required for a pre-set overall accuracy will vary. It is, therefore, desirable to be able to alter the iteration timestep during modelling, so that the accuracy requirements of the problem may be met with the minimum of computing resources. Fundamental to any automatic timestepping procedure is a rapidly implemented method of estimating the error at each iteration. Here we describe two methods of error estimation together with a method of implementing timestep changes.
ERROR ESTIMATION
2.1. The difference method
In the thermal case, the diffusion equation, in two dimensions, is d2T a2T SpdT -+ -= --ax2 ay2 K at where K = thermal conductivity, S = specific heat, and p = density. dimensions, is TLM algorithms solve equations of the form of the telegrapher's equation which, in two t Now with