Making Laasonen More Accurate

The two methods are BDF and extrapolation. Both methods are used for the numerical solution of odes and are described in Chap. 4. The extension to the solution of pdes is most easily understood if the pde is semidiscretised that is, if we only discretise the right-hand side of the diffusion equation, thus producing a set of odes. This is the Method of Lines or MOL. Once we have such a set, as seen in (8.9), the methods for systems of odes can be applied, after adding boundary conditions. 

The BDF method has been described. One starts with the system such as (8.9), and goes on from there as described. This was first suggested by Richt-myer in 1957 [470J, who suggested the three-point variant, and was first used in electrochemistry by Mocak and Feldberg [402] and later refined to variable time intervals by Feldberg and Goldstein [236]. These workers call it FIRM, 

There have been attempts to improve the performance of BDF, which is normally limited by the second-order (in the spatial interval H) discretisation of the spatial derivative. Higher-order spatial second derivatives have been tried out in connection with BDF [152,154], They can only work as intended if a high-order start is used, such as the KW start as described in Sect. 4.8.1. This start was not found to be efficient in [154], but it may be that a technique other than the one used there, such as Numerov (see Chap. 9), which does not produce banded matrices, will make the use of KW efficient and thus interesting. For this reason, the KW start is described below. 

it is worthwhile detailing the implementation of BDF itself, ignoring startup for the moment. We choose 3-point BDF. Based on (4.28) given on page 57 for odes, the diffusion (8.8) is discretised at index i as 

Now for the KW start for BDF. The description in [154] will be followed here. First of all, (8.47) is rewritten in ode form for the whole system, replacing the left-hand side by the time derivative and the right-hand side by the general matrix form 

Corresponding higher-order forms can now be constructed using the examples in Chap. 4, Sect. 4.10. In the present context, the matrix equations get rather large for larger k. If there are N unknowns across the spatial dimension, then the matrix equation will be (fc — 1)1V x (fc — l)N. So the method might be suitable only for smallish N. 

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