Direct Application of an Arbitrary Grid

We have seen from the above that, in some way or other, we choose the value of Hi X i. We also have a maximum value X / along X, which depends 

The easy alternative is to set a. One develops a feeling for what value might be a good one. Having set this value and knowing that of the above (7.12). .. (7.14) yield N. 

Alternatively, one might want to set Xi and N and find an a value that provides these. This is a little harder. Substituting for Yjv from (7.14) in (7.12) and combining (7.13) with the result, we obtain the function 

Lastly, it is possible also to set a, Xl and N, and to use them to find 6Y and thereby X1. If it is done on a calculator beforehand, one sees what value results, before committing the chosen parameters to a simulation run. 

A stretched stack of boxes was used by Feldberg [231] for the box-method, to be described in Chap. 9. Pao and Dougherty [433] developed the same idea 

A stretched stack of boxes was used by Seeber and Stefani and by Feldberg [7, 8] for the box-method, to be described in Chap. 9. Pao and Dougherty [17] developed the same idea (and stretching function) in 1969, in the context of fluid dynamic simulations. This is the simple placement of points at increasing intervals, in some suitable point distribution or stretching function, and discretisation of the second derivative of concentration along X on that unequal grid. 

There are various ways of specifying the stretched point placement. The current favourite appears to be the exponentially expanding sequence of intervals H along [7, 8], 

As will be seen below, the way stretched intervals are used here is that a set of positions in X are specified. We must therefore convert the intervals formula above (7.17) to one in terms of X. For any N 0, 

The drawback of this point sequence (and most others except a sequence of equal intervals) is that the three-point approximation to the second derivative with respect to X is then a first-order approximation, as was mentioned in Chap. 3, Sect. 3.8. The use of more than three points is thus indicated, and such approximations are described in Chaps. 3 and 9, and some formulas are given in Appendix A. 

There is one unequal sequence of points for which the second derivative, when applied directly to the points, retains the second-order nature of an even point spacing. This was found by Sundqvist and Veronis [20] in 1970. Their stretching function was 

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