Space discretisation

When a stochastic model is described by a continuous polystochastic process, the numerical transposition can be derived by the classical procedure that change the derivates to their discrete numerical expressions related with a space discretisation of the variables. An indirect method can be used with the recursion equations, which give the links between the elementary states of the process. 

The constituent subprocesses of discretisation consist of mesh generation (determining the types of mesh elements), space discretisation and time discretisation. 

Depending on the space discretisation techniques used, the set of equations to be solved may be different, but for FD- and FV- based methods, the discretisation results in a set of linear or non-linear algebraic equations. These depend on the nature of these partial differential equations and how they are derived. For linear equations, it is well known that a Gauss elimination method can be used as a basic method to solve them. Further details of the Gauss method can be found in [60],... 

Ui Vi Ci) = Uijrnax,Vijrnax,Cijrnax) For the space discretisation we have used standard linear finite elements on a uniform triangular grid. 

Irrespective of the type of extended system we are interested in we impose periodic boundary conditions in position space - "the large period" BK. Such conditions imply a discretisation of momentum and reciprocal space 27] which means that integrations are replaced by summations ... 

For computational reasons, it is necessary to discretise the space in such a way that one considers only a finite number of z-coordinates for the segments of the molecules to take positions. Again, the coarse graining is justified as long as the relevant phenomena do not take place on a smaller length scale. Here we consider lattices with sites that fit united atoms, e.g. CH2 units. The... 

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. 

Transformation (7.3) leads to the new diffusion (7.5) in Y-space. Although it is fairly obvious how the new right-hand side is discretised, for completeness, this will be described here. 

Instead of a number of sample points in X, we now have a number of equally spaced points along the new coordinate Y with a spacing of 6Y. Without considering which simulation algorithm is to be used, we discretise the new (7.5) at the point Y.j as follows ... 

As for the choice between direct discretisation on an arbitrarily spaced grid or the formulae for the semi-transformed or the transformed diffusion equation, the present author now inclines towards the first of these. Formulae for the derivatives on arbitrarily spaced points are given in Chap. 3 and Appendix A, and the general subroutine U DERIV is referred to in Appendix C. 

Just as space can be divided into unequally spaced intervals, so might time also be unevenly divided. As with spatial intervals, there is the choice between discretising on an uneven time grid or using a transformation to a new time scale. Since, except for BDF methods, one usually differentiates with respect to time using only two time points (levels), transformation does not make sense here. 

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... 

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,... 

In general, the finding of Rudolph [478], that in one-dimensional simulations, direct discretisation on an unequally spaced grid, rather than equal spacing on a transformed grid, is best, does not appear to apply to UME simulations. Gavaghan made a very thorough study of UMDE simulations... 

One has the choice between applying finite difference discretisation either directly to a grid of points in the cylindrical (R, Z) space, or to a transformed space. In one dimension, it has been found [478] that direct discretisation without transformation is better. In the case of 2D simulations where edge effects are seen, this is not the case, and transformation is better. Both approaches are described here. 

We wish to simulate by discretising on an equally spaced grid in the transformed space, and this grid should place points optimally in the original (R, Z) space. That is, they should be closely spaced near the disk edge,... 

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... 

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. 

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. 

It is important to notice that the univocity conditions must adequately correspond to the process reality. Concerning the numerical discretisation of each variable space, model (4.146) gives the following assembly of numerical relations ... 

In this woric, discretisation of both space and time derivatives was executed, based on either central finite difference (CFD) or orthogonal collocation cm finite elements (OCFE) discretisation in the spatial domain and backward finite difference (BFD) discretisation in the time domain. 

In the finite difference method, the derivatives dfl/dt, dfl/dx and d2d/dx2, which appear in the heat conduction equation and the boundary conditions, are replaced by difference quotients. This discretisation transforms the differential equation into a finite difference equation whose solution approximates the solution of the differential equation at discrete points which form a grid in space and time. A reduction in the mesh size increases the number of grid points and therefore the accuracy of the approximation, although this does of course increase the computation demands. Applying a finite difference method one has therefore to make a compromise between accuracy and computation time. 

---