The Discretisation

The discussion will be restricted to the point method and to the onedimensional case. We will now work in normalised variables, see Sect. 2.3. We then have concentration points Cq, C, ... C/y, Cqv+i, at the locations 

At any point with index i, that is at X = iH, the diffusion (1.1) is discretised on the left-hand side in the Euler manner (4.4, or in other words the forward difference formula 3.1) and on the right-hand side with the central three-point approximation (3.41), giving for the iteration going from time T to the next time T + 8T, 

Dieter Britz Digital Simulation in Electrochemistry, Lect. Notes Phys. 666, 73-83 (2005) www.springerHDk.ccm -c- Springer-Verlag Berlin Heidelberg 2005 

The notation is now simplified by always assuming that we are at time T and are going to time T + ST and writing C(T + ST) as C. The equation then rearranges to 

The outer point CV+i is normally equal to the bulk initial value, and thus equal to unity, since concentrations have been normalised by the bulk value (in cases involving more than one diffusing species, their respective initial bulk values, normalised by that of the chosen main species). The value of Co is a little more complicated to set. It depends on the experiment to be simulated, and for simplicity at this point the discussion will be postponed to a later section in this chapter. Assume that we know the value Co- Then we need only go through all concentrations Ct. .. CN, applying formula (5.2), and obtain the new row of values C[. .. C N. 

The discussion will be restricted to the point method and to the one-dimensional case. We will now work in normalised variables, see Sect. 2.3. We then have concentration points Co, Ci. Cjv, Cat+i, at the locations X = 0,H. NH, (N + l)H, H being the interval in X, see Fig. 5.1. The end points at X = 0 and X = N + )H are boundary points with concentrations Co and Cjv+i, respectively. It is the concentrations between these, that are subject to diffusional changes, as follows. 

Stmtwolf, Digital Simulation in Electrochemistry, Monographs 

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Whilst finite-element modelling of gap junctions occurs at a sub-cellular level, these models do not consider the operation of intact organs. Conversely, in models of the complete heart the discretisation is usually on a milhmetre scale. However, the cochlea (see Figure 9.3) is already being simulated on a 0.01 mm, or cellular, scale. Although cochlear malfunction is not hfe threatening, damage to it does adversely affect the ability of almost 1000000000 people to communicate. 

The coefficients are used to compute the derivatives, but can also be useful in the discretisation of derivative boundary conditions or in the setting up of discretisation matrices in some problems. 

In some cases, for example electrochemical pdes with derivative boundary conditions, the discretisation process for both the pde and the boundary conditions leads to a mix of a differential equation system and one or more plain algebraic equations. They might be, for example, equations of the form... 

If several species are involved (in this case there is the product prod, but we are not interested in it), the equations are extended in an obvious manner, apart from some tricks to be seen in a later chapter in connection with implicit methods. This is one of the attractive aspects of method EX. If the her is second order, there will be a term in C in the discrete equation, and it will present no problem in the discretisation step [146]. 

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,... 

The Feldberg approach to digital simulation [229] uses a somewhat different method of discretisation, and the method is alive and well it is, for example, the basis for the commercial program DigiSim [482], It begins with Fick s first diffusion equation, using fluxes between boxes or finite volumes, rather than concentrations at points in the discretisation process (see below). 

This expanding box strategy is mathematically equivalent to the transformation from X into Y as described for point positions in Chap. 7, (7.3), as is shown in Appendix B. Its implementation in the discretisation process is however different. 

This early paper was followed by another one in 1990 by Kimble and White [338], now applying the method to a diffusion problem, and using 5-point approximations in both directions. As before, the problem was cast into a block-matrix, but because of the 5 points used for the discretisations, this was block-pentadiagonal. For most node points in the figure, the 5-point approximations yield the following computational molecule or stencil. 

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,... 

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... 

We are now ready to apply the discretisations, but must decide on the vector of unknown concentrations at all the grid points in Fig. 12.3. It is convenient to include even the boundary points (but not those at j = —1, which serve only as fictitious points), setting these to known values in the large linear system to be generated. Thus we note that the total number N of unknowns is given by... 

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... 

To make the discretisation process more visual, consider any position (i, j) in the grid. There are a total of 7 points around and including this central point, and each of them has its own A -value, mapped from its indices. It looks like this ... 

We can now put the discretisations together, still focussing only on the right-hand side of (12.17). Adding up the individual discretisations (12.37), (12.38) and (12.39), we can express the total (semi) discretisation as... 

The aim of a simulation is to approximate the underlying exact solution as accurately as possible, in a minimum of computer time. Solution is achieved by some discrete formula, which has truncation errors, due to neglect of some (higher) Taylor terms in the discretisation formulae. These errors must become smaller as we make the intervals both in time and space smaller and the errors must, at least, not grow in the course of a number of steps. This property is called convergence. In the limit, as 6T and 6X (that is. H) approach zero, the errors must also do so. In order for this to happen, two conditions must hold. The first is that the discretisation expression used must be consistent with the differentia] equation it approximates. The second is that the expression must be stable. This means that an error in the solution at a given step is not amplified by subsequent steps. These two issues will be examined separately. 

In the present context we have two intervals ST in time, and II in space. A given discretisation is said to be consistent if, as both of these intervals approach zero, the discretisation approaches the pde it is meant to approximate. Take the simple explicit discretisation on equal intervals, in (5.2), which we rearrange into the form... 

The methods considered above all show purely real eigenvalues. One method for which they are complex is BDF. For 3-point BDF, the discretisation is... 

The proof is given for a general ode. It also applies to a system of odes and thus to the system of equations resulting from the discretisation of a pde. Let the equation to be solved be... 

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. 

Numeric dispersion can be eliminated largely by a high-resolution discretisation. The Grid-Peclet number helps for the definition of the cell size. Pinder and Gray (1977) recommend the Pe to be < 2. The high resolution discretisation, however, leads to extremely long computing times. Additionally the stability of the numeric finite-differences method is influenced by the discretisation of time. The Courant number (Eq. 104) is a criterion, so that the transport of a particle is calculated within at least one time interval per cell. 

With the finite-elements method the discretisation is more flexible, although, as with the finite-differences method, numeric dispersion and oscillation effects can... 

The discretisation shall be carried out in elements of 10 m length. The connection of the immobile cells to the mobile cells is done by a box for each cell (Fig. 45) and the exchange between mobile and immobile cells by the means of a 1st order reaction (for theory see chapter 1.3.3.3.1). Present the concentrations of the elements U, Fe, Al, and S at the pumping well over a period of 200 days. 

Fig. 2.43 Grid for the discretisation of the heat conduction equation (2.236) and to illustrate the finite difference equation (2.240)...

The discretisation error goes to zero with a reduction in the mesh size Ax or At. 

The writing of O (Ax2) indicates that the discretisation error is proportional to Ax2 and therefore by reducing the mesh size the error approaches zero with the square of the mesh size. The first derivative with respect to time is replaced by the relatively inaccurate forward difference quotient... 

The derivative is formed in the outward normal direction, and a and can be dependent on time. For the discretisation of (2.253) it is most convenient if the boundary coincides with a grid line, Fig. 2.45, as the boundary temperature which appears in (2.253) can immediately be used in the difference formula. The replacement of the derivative d d/dn by the central difference quotient requires grid points outside the body, namely the temperatures i9k or ()k,, which, in conjunction with the boundary condition, can be eliminated from the difference equations. 

The discretisation error of this equation is 0(Ax) and not 0(Ax2) as for (2.256), because the approximation (2.260) was used. The use of (2.262) therefore leads us to expect larger errors than those with (2.256). On the other hand, the stability behaviour is better. Instead of (2.259) the condition... 

The establishment of difference equations is based on the discretisation of the self-adjoint differential operator... 

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