## J

(a) Variation of u with z

(b) Schematic form of equation

Fig. 10.13 Explicit recurrence formula: treatment for an impermeable boundary.

derivatives) and rounding-off errors (due to working to only a certain number of decimal places). The size of the space increment, Az, affects both these errors but in different ways: the smaller Az is, the less the truncation error that arises but the greater the round-off error tends to become.

The value of r is also important. For stability r must not be greater than 0.5 and, for minimum truncation errors, should be 1/6; the usual practice is to take r as near as possible to 0.5. This restriction means that the time interval must be short and a considerable number of iterations become necessary to obtain the solution for a large time interval. With present software this is not a problem, but if necessary use can be made of either the implicit finite difference equation (Crank and Nicolson, 1947) or the relaxation method (Leibmann, 1955).

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