Discretisation first spatial derivative

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,... 

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. 

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... 

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... 

Spatial discretisations on unequally spaced points are best done using multi-point stencils, for example five-point. However, in order to keep things simple, three-point approximations are used in what follows. The symbols like i)k refer to three fi coefficients k = 1,2,3 pertaining to point i, used to approximate a first derivative at point i, and similarly for the a coefficients. These are all precomputed using the Fornberg algorithm [63], implemented in the subroutines forn and fornberg, described in Appendix E. 

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