Discretisation second spatial derivative

As mentioned above, Rudolph [478] pointed out that this discretisation yields very poor values and ultimately to poor simulation performance, compared to direct discretisation on an uneven grid, see below. Tests show that particularly at small X values, near the electrode where the greatest changes occur, the second spatial derivatives as seen in (7.7) are approximated very poorly. Rudolph [479,480] and Bieniasz [107] showed that if what we might call the semi-transformed (7.1) is used, rather than the hilly transformed equation, this problem is eliminated. Doing this in a consistent manner, and assuming general transformation functions f(X) and g(Y), we can write for the ith point the approximation... 

First, the discretisation of the second, spatial derivative of concentration will be reiterated in a general form that can then be built into the methods to follow. For the three concentrations grouped around the one at the point Xi, we can write the general linear expression,... 

It was shown in Chap. 7 that the three-point second spatial derivative on an unequally spaced grid, leading to (8.1) with the coefficients defined in (8.3), can be improved with relatively small effort to an asymmetric four-point, formula, spanning the indices i — 1, i, % f 1, % I 2, with the second derivative referred to the point at index i. The diffusion equation is then semi-discretised to... 

Kimble and White were aware that leapfrog methods are unstable and simply remark that this did not seem to apply to their method. Also, they mention the use of 5 points for all approximations but their table of discretisations shows that they used 6 points at the edges for the spatial second derivative. This is no doubt because, as Collatz already mentions in 1960 [170], the asymmetric 5-point second derivative is only third-order, while a 6-point formula is fourth-order, like the symmetrical 5-point ones used in the bulk of the grid. So, for the second spatial derivative at index i = 1, the form 2/2(6) was used, and the reverse, form 2/5 (6) at i = N. 

In this scheme, the temporal derivative is formed by the central (second-order ) difference between the upper and lower points, the second spatial derivative being approximated as usual. This makes the discretisation at the index i in space,... 

Unequal intervals Chap. 7. These are essential for most programs. The second spatial derivative requires four points if second-order is wanted (and is recommended). With four-point discretisation, an efficient extended Thomas algorithm can be used, obviating the need for a sparse solver. Very few points can then be used across the concentration profile. For two-dimensional simulations, direct three-point discretisation on the unequally spaced grid was shown to be comparable with using transformation and discretisation in transformed space. 

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. 

In digital simulation, when discretising the diffusion equation, we have a first derivative with respect to time, and one or more second derivatives with respect to the space coordinates sometimes also spatial first derivatives. Efficient simuiation methods will always strive to maximise the orders. 

The time derivative is still a central difference but the spatial second derivative now leaves out the central point, substituting for it the mean of the past and future points. Thus, the discretisation is... 

Firstly, the discretisation itself is described. We restrict the discussion to the BI time integration, in order to focus on the spatial discretisations. The program UMDE DIRECT in fact uses BI as the first step, then three-point BDF, which produces second-order accuracy with respect to ST, this being the rational BDF startup described in Chap. 4, page 59. Take a point away from the boundaries, indices i (for Z) and j (for R). The discretisation at concentration (r j of the pde (12.17) has three derivative terms, all to be discretised using four-point formulas. The coefficients can be precalculated. For the row along Z, there are, for each 0 < Z < Zmox, that is, 0 < i < n-z, four coefficients for the approximations... 

Higher-order methods Chap. 9, Sect. 9.2.2 for multipoint discretisations. The four-point variant with unequal intervals is probably optimal the system can be solved using an extended Thomas algorithm without difficulty. Numerov methods (Sect. 9.2.7) can achieve higher orders with only three-point approximations to the spatial second derivative. They are not trivial to program. 

This subroutine is a general routine for computing the first or second derivative on a number n of points, referred to the ith one among that number, on an arbitrarily spaced grid of points. The derivatives are computed as a linear sum of terms, and the coefficients in that sum are also passed back, for use, for example, in the discretisation of boundary conditions or the spatial second derivative. The number of points is in principle unrestricted, but the routine will fail for values n > 12, where the accuracy abruptly drops. A value, in any case, exceeding about 8, is perhaps impractical. This routine can be used instead of the algebraic expressions shown in Chap. 7, or if n values greater than 4 are required. 

Britz D (2003) Higher-order spatial discretisations in digital simulations. Algorithm for any multi-point first- or second derivative on an arbitrarily spaced grid. Electrochem Commun 5 195-198... 

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