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As with any numerical method, convergence tests must be performed. In the present case, these are necessary in order to determine that the results are independent of the intervals Sz and 6t. For the characteristic dimensions of interest in the following problems, namely nanostructures (~100 1000 A) and macroscopic annealing times (~100 1000 s), 6z = 1 A and 6t = 0.01 s were found to be satisfactory. From a computational viewpoint, the technique is useful since, in general, it is anticipated that the numerical complexity scales linearly with the physical complexity. In actuality, it has been found that for the series of problems discussed below, the number of spatial points required for the calculation is independent of the variability in the initial concentration profile. Increasingly larger microstructures, i.e. superlattices with more and more periods, do require more points (i.e. 6z remains constant), but the computational effort required is proportional to the number of points. For a given structure, but with an increasingly complex dependence of the diffusion coefficient, e.g. V having a stronger functional dependence on the concentration x, then smaller time intervals 6t are required. Again, however, halving the time interval merely doubles the necessary computational effort. Having therefore established a numerical method for simulating diffusion and after validating it by comparison to the accepted analytical form for a special case, it is now an appropriate time to demonstrate the solution's versatility by visiting, in turn, the examples (i-iv) listed in Section 4.2.

The objective of this chapter has been to summarize the main elements of program construction, using a concrete example as illustration. A brief summary is that there are three essential ingredients: the 'what', the 'why' and the 'how'. The 'what' is the program's specification. A specification acts as a contract between the client, who commissions the software, and the programmer, who implements the software; it says what the program should compute. The 'how' is the implementation itself. It consists of a number of instructions that are executed by a computer. The implementation thus prescribes how to meet the specification.

Figure 4.4 Comparison between the numerical solution (solid symbols) and the standard error-function solution (continuous line) for the concentration profile x(z) of a 200 A Gao. 9Alo.1As/200 A GaAs /200 A Gao.9Alo.i As single quantum well, after 100 s of diffusion described by a constant diffusion coefficientDo=10 A2s-1

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The 'why' is the justification that the implementation does indeed meet its specification. It documents the program by explaining the function of the instructions in the implementation; it says why the program is as it is. A precise specification is essential to the construction of reliable programs; without one, there is simply no basis for verifying that the client's requirements have been met. Also, because it is to be executed by an unforgiving machine, the implementation needs to be precise and unambiguous. Finally, the justification that the program meets its specification needs to be precise, in order to have complete confidence in the correctness of the implementation. In this chapter, we have foregone precision for the sake of illustration. This lack of precision means that we cannot truly claim to have developed a correct implementation of the given specification. The problem lies in the use of an imprecise language (English) to discuss the program; at times you have been expected to interpret the statements generously. (For example, the way in which the lower and upper decks, or the left and right decks, are combined into one was never precisely stated, although it is crucial to keeping the original deck intact.) In later chapters, we rectify the problem. We introduce a simple language for program specification, and a separate language for program implementations. We also relate the two, precisely and concisely.

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