Current approximation hermitian

High-Order Compact (Hermitian) Current Approximation... 

As already described in some detail in Chap. 3, a one-sided first derivative such as the current approximation G can be raised to higher-order by a Hermitian scheme, as introduced by Bieniasz [108], This can then be used both to obtain better current approximations, and also in those cases where G enters a boundary condition. For the simpler case of the current approximation on a concentration grid already calculated, see the relevant Sect. 3.6 in Chap. 3. Here we need to go into some detail on the boundary conditions application. 

There are simulation cases (for example using unequal intervals) where it is desirable to use a two-point approximation for G, both for the evaluation of a current, and as part of the boundary conditions. In that case, an improvement over the normally first-order two-point approximation is welcomed, and Hermitian formulae can achieve this. Two cases of such schemes are now described that of controlled current and that of an irreversible reaction, as described in Chap. 6, Sect. 6.2.2, using the single-species case treated in that section, for simplicity. The reader will be able to extend the treatment to more species and other cases, perhaps with the help of Bieniasz seminal work on this subject [108]. Both the 2(2) and 2(3) forms are given. It is assumed that we have arrived at the reduced didiagonal system (6.3) and have done the u-v calculation (here, only v and iq are needed). 

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