Backwards Implicit BI

Another possibility is to let the same derivative approximation pertain to the next time this is the backward implicit (BI) method  

This method seems at first sight unpromising, because of its low error order, the same as that for Euler. However, it has some very useful stability properties (see later) and forms the basis for several high-order methods, as will be seen. 

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Finite difference methods have been used bpth to test the assumptions made in the derivation of eqn. (27) under the Leveque approximation [35] and to solve electrochemical diffusion-kinetic problems with the full parabolic profile [36-38]. The suitability of the various finite difference methods commonly encountered has been thoroughly investigated by Anderson and Moldoveanu [37], who concluded that the backward implicit (BI) method is to be preferred to either the simple explicit method [39] or the Crank-Nichol-son implicit method [40]. 

Time integration methods more advanced than the one considered in this book (backward implicit, BI) can also be employed. Indeed, the Crank-Nicholson and high-order extrapolation methods [6] have proven to enable the reduction of the number of timesteps (and even improve the accuracy of the simulation) with respect to BI [4]. 

The explicit (Euler) method described above has this stability limitation. There are other methods that do not. One of them (reverting again to the ode 21) is the backward difference (or backward implicit, BI) formula ... 

Then, the new value of B is used in the isotherm equation to recalculate Q. This is simple, but has the drawback of poor accuracy and the limit on the A factor. Clearly, an implicit method is preferable, such as BI with extrapolation. The diffusion part of the whole set of equations will depend on the placing of the points in space, as described in Chap. 8, for example using the general three-point equation (8.8) on page 147, or a multi-point form such as (8.31), page 151. These lead to the usual system as (8.11) or its multi-point relative (8.33) on pages 147 and 151. The first step is to do the backward Thomas scan as described in that chapter, and to apply the M-v procedure, resulting in a set of linear expressions for the first n (as yet unknown) concentration values, in terms of Cq,... 

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