Physical Constraints in Numerical Calculations of Diffusion

The physical behaviour of the diffusion equation is examined, and shown to be a consequence of appropriate mathematical properties of the diffusion operator. Amongst these, the familiar decay of extrema, a consequence of the maximum principle, is is given particular attention. The development of spatial and temporal differencing to preserve this property, to be called extremal, yields solutions which preserve positivity and converge uniformly to the steady state. The general construction of extremal algorithms is described for use in a two-level system. The use of weights to improve the accuracy of temporal integration is discussed.