Laasonen method extrapolation

This is an example of a Cottrell simulation using second-order extrapolation based on the BI (Laasonen) method and unequal intervals. Three-point spatial discretisation is used here. 

This program is again a Cottrell simulation using second-order extrapolation based on the Bl (Laasonen) method and unequal intervals, but in contrast with the above program C0TT EXTRAP, this one makes use of the four-point spatial derivative approximation, and the GU-function. It performs a little better than the above program, at little extra programming effort. 

The alternative to BDF, that still makes use of the pleasant properties of the Laasonen or BI method, is extrapolation. When one says that discretization (27) is first order with respect to St, one means that the error e, can be expressed as a polynomial like... 

As with BDE, the simpler second-order scheme appears about optimal. This method also shows the same smooth and damped error response of Laasonen, with the accuracy of CN. The drawback is that for every step, several calculations must be performed—in the case of second-order extrapolation, three in all (see Sect. 4.9). This also implies an extra concentration array, for the final application of the formula, for example the vector equivalent of (4.31), requiring the result of the first, whole step, and then the result of the two half-steps. Discretisation for extrapolation is the same as for Laasonen [coefficients as in (8.12)], but using two different values of ST and therefore two coefficient matrices. These can of course be precomputed if the coefficients are constant over the simulation period, so this is not a great problem. There are example programs using extrapolation (COTT EXTRAP and C0TT EXTRAP4) referred to in Appendix E. 

The IDA generator-collector system has been simulated using the Laasonen (BI) scheme [374], explicit FD [375, 376], hopscotch and conformal mapping [220], the finite analytic numerical method [152], extrapolation using expanding space intervals [332] and ADI with expanding space intervals [348, 377]. Commercial FEM software packages Flux-Expert [324] and COMSOL Multiphysics [378] have also been employed. 

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