Discretisation errors

The derivatives which appear in (2.236) are replaced by difference quotients, whereby a discretisation error has to be taken into account... 

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... 

Its discretisation error only approaches zero proportionally to At. However with (2.239) a numerically simple, explicit finite difference formula is obtained. Putting (2.238) and (2.239) into the differential equation (2.236) gives, after a simple rearrangement, the finite difference equation... 

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... 

If the larger discretisation error of (2.262) is to be avoided, the boundary temperature has to be replaced by a more accurate expression than the simple arithmetic mean according to (2.260). A parabolic curve through the three temperatures o, i and 2 gives... 

The discretisation of the heat conduction equation can also be undertaken for three-dimensional temperature fields, and this is left to the reader to attempt. The stability condition (2.304) is tightened for the explicit difference formula which means time steps even smaller than those for planar problems. The system of equations of the implicit difference method cannot be solved by applying the ADIP-method, because it is unstable in three dimensions. Instead a similar method introduced by J. Douglas and H.H. Rachford [2.71], [2.72], is used, that is stable and still leads to tridiagonal systems. Unfortunately the discretisation error using this method is greater than that from ADIP, see also [2.53]. 

This is an approximate equation as each small, but finite block has only one discrete temperature associated with it and because only the heat conduction between immediate neighbours has been considered. As we can show with the use of the discretisation equations (2.299) and (2.300) for the second derivatives, we also obtain (2.308) by the usual discretisation of the differential equation (2.306). The discretisation error in this case is 0(Ax2) it decreases to zero with the square of the mesh size (= block width). 

The last part of the process is now to solve (integrate) the equation system 5.129. We could use the simple Runge-Kutta method, probably going to a fourth-order scheme since we (hopefully) are not limited here by the second-order discretisation error inherent in system 5.13. it is more common to employ a more sophisticated technique. Whiting and Carr (1977) suggest Hamming s modified predictor-corrector method, Villadsen and Michelsen (1978) that of Caillaud and Padmanabhan (1971) (and provide the actual subroutines). There are other methods. The criterion will always... 

In this section, error sources, cancelling and fudges are described. 6.3.1 Discretisation errors... 

Rudolph M (2004) Digital simulations on unequally spaced grids. Part 3. Attaining exponential convergence for the discretisation error of the flux as a new strategy in digital simulations of electrochemical experiments. 1 Electroanal Chem 571 289-307... 

As with BDF, the simpler second-order scheme appears about optimal. This method also shows the same smooth and damped error response of Laa-sonen, 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 Laa-sonen (coefficients as in (8.12)), but using two different values of 6T. There are example programs using extrapolation (COTT EXTRAP and C0TT EXTRAP4) referred to in Appendix C. 

We can also choose not to linearise the nonlinear term by an approximation, in which case we do not run the (minimal) risk of adding errors to the simulation by the linearising approximation. The same example as used above (8.57) and again choosing CN as the method, the dynamic (8.58) is discretised 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 case of the ultramicroelectrodes such as the disk electrode, it is necessary to integrate over the surface, and sometimes there will be unequally spaced points along the surface, as for example, in direct discretisation on an unequal grid in the example program UME DIRECT. As mentioned in Chap. 12, it is found that due to the errors in the computed concentration values, the local fluxes are so inaccurate that any integration method better than the simple trapezium method is not justified. The routine U TRAP is thus recommended here. It integrates local current densities, precalculated by using the above routine U DERIV. 

Equation 4 was discretised by a 5-point central difference formula and thereafter first-order differential equations 1, 2, 4 and 6 were solved by a backward difference method. Apparent reaction rate was solved by summing the average rates of each discretisation piece of equation 4. The reactor model was integrated in a FLOWBAT flowsheet simulator [12], which included a databank of thermodynamic properties as well as VLE calculation procedures and mathematical solvers. The parameter estimation was performed by minimising the sum of squares for errors in the mole fractions of naphthalene, tetralin and the sum of decalins. Octalins were excluded from the estimation because their content was low (<0.15 mol-%). Optimisation was done by the method of Levenberg-Marquard. 

In the present work Gallerkin s method of weighted residuals is used to derive the weak form of the equilibrium equations. Hence, the first step towards finite element discretisation of the governing equations is the definition of shape functions for the domain variables, i.e. displacement, pore water pressure and pore air pressure. Introducing these shape functions into equations 13, 14 and 15 the governing equations are approximated with a certain accuracy. The approximation errors, termed... 

One problem with all these discretisation formulae is that they provide absolutely no information about possible errors. Often, this may be academic - we usually "dry-run" our serious simulations on systems that have known solutions, to be reasonably sure they are working - but it may be of interest. 

The coefficients we have used in the above are those for the so-called classical schemes they are chosen so as to minimise the error but others have been derived. These classical schemes (coefficients) will be used in the book. The higher-order formulae give correspondingly smaller errors (of higher order in at). As will be seen in Chapt. 6, there is seldom a need to go beyond second order because of the second-order error (in h) made in the discretisation of the diffusion equation which, in the present context, limits the accuracy of the derivative /(u). 

We assume, for a start, the simple diffusion equation 5.12. We have seen in Sect. 5.1, that the normal explicit method, with its forward-difference discretisation of 8c/3t performs rather poorly, with an error of 0(6t). The discrete expression for the second derivative (right-hand side of Eq. 5.12) is better, with its error of 0(h ). Let us now imagine a time t+ig6t at this time, the discretisation... 

In rewriting the diffusion equation (with possible extra terms) as a set of discrete approximations, we make a number of errors. The discretisation with EX of the left-hand side, that is... 

Closely connected with error cancelling, fudge factors have enjoyed some popularity in digital simulation and to some extent still do so. These are sometimes arbitrary factors thrown in to "improve" the results and sometimes incorrect derivations of the discretisations. A good example of the first sort is the "correction factor" used by Prater and Bard (1970) who - to be fair - were not the first. By this means, they were able to simulate a rotating ring-disk system with only 50 time steps and they do warn the reader that, for other 6T, different correction factors may be needed. 

First of all, these 0(6T ) errors refer to a single 6T step, whereas in the simulation in the T-range 0 < T < 1 we take 6T steps. Generally, this reduces the error order by one. Tests show that it is in fact the discretisation of the second derivative which limits the accuracy, hence the 0(H ) error for all methods. We might say that, in view of this, we are lucky that the better methods (CN, RKI) give better results, since they suffer from the same error source. 

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