Propagational inadequacy

Both the LR and RL variants, despite being explicit, are said to be stable for all A values, which is a great advantage. Also, the method does not share with DuFort-Frankel and hopscotch the propagational inadequacy problem [232] mentioned above because both variants amount to a recursive algorithm, each newly calculated element carrying with it some component from all previously calculated elements. 

Hopscotch Chap. 9, Sect. 9.2.5. Stable and explicit but has the problem of propagational inadequacy , so the ability to use large A values is lost. Nevertheless, hopscotch continues to be used. 

Feldberg SW (1987) Propagational inadequacy of the hopscotch finite difference algorithm the enhancement of performance when used with an exponentially expanding grid for simulation of electrochemical diffusion problems. J Electroanal Chem 222 101. 

One method apparently still in some favour, is hopscotch, already mentioned in Chap. 9, extended to two-dimensional problems by Gourlay [215], as did Evans [101] in the same year. Despite the indication by Feldberg [109] and Carnahan et al. [108] that hopscotch suffers from propagational inadequacy (Feldberg s term), it has continued to be used by electrochemists [105-107, 148, 216-222]. Danaee and Evans [223] described a composite method using points and blocks, that may have fixed the propagation problem but this has not been tried in electrochemical work. Hopscotch is in some ways similar to Crank-Nicolson and like that method, has an oscillatory error response, as is seen in [223] and [218], among others. It has been described in Chap. 9. 

Since most types of experimental inadequacy or incompetence produce more ethanol in the hydrolysate than could be derived from fert.-oxonium ions, one must conclude from this evidence that the propagating species is a secondary oxonium ion. It follows necessarily that the ring-expansion mechanism is the best representation of the propagation reaction, that tert.-oxonium ions do not play an essential role in the polymerisation of DCA by perchloric acid, and that therefore this part of the old controversy appears now to be settled. 

Fig. 23 Illustration of the nonlinearity of the accumulated damage and inadequacy of the linear approach for extrapolating crack propagation for future operating periods.

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