Discretisation explicit method

There are now several choices of method. The simplest may be the explicit method. Using this, one starts at time T, where we know all values, and use them to proceed to the new time T + 6T. First, one recalculates all G.j, i 1. .. A. Parallel with this, from the value of G, one calculates a new 6. Discretising the second equation of the set (10.3) and expanding G as usual as an n-point approximation leads to... 

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... 

The restriction on the step size (2.304) due to the stability condition for the explicit difference method can be avoided by using an implicit method. This means that (2.298) is discretised at time tk+1 and the backward difference quotient is used to replace the time derivative. With... 

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]. 

In Chapter 3, we chose to use an implicit method of solution, as opposed to an explicit one for reasons of stability and simulation efHciency (despite the greater complexity of the implicit method). The implicit discretisation... 

Lapidus and Finder (1982) describe a method which meets our requirements the alternating directions implicit or ADI method of Feaceman and Rachford (1955). Consider Fig. 8.2, and Eq. 8.32 to be simulated. At time T, we work all rows, using fully implicit discretisation (in the Laasonen 1949 sense) in the X-direction but explicit discretisation in the Y-direction. Then, at a given (i,j) coordinate, assuming again 6X = SY = H,... 

So far, nothing has been said about boundary conditions. The ADI method can, in fact, be thought of as a type of CN, on average, as every point is recomputed from a mixture of explicit and Laasonen-implicit discretisations. It is to be expected that, just as in one dimension when using the CN method, derivative boundary conditions can cause trouble (Britz and Thomsen 1987). The same remedy can be applied here using implicit expressions for the boundary concentration. As a concrete example, take the microdisk. Fig. 8.3. It is symmetrical about the axis... 

Finite Volume Methods The finite volume method, when the permeabihty tensor is diagonal in the selected coordinate system, approximates the pressure and saturation functions as piecewise constant in each grid block. The flux components are assumed constant in their related half-cells. Thus when two cells are joined by a face, the related component of flux is assumed to be the same each side of the face. The balance laws are invoked separately on each grid block, and are discretised in time either by an explicit or fully implicit first order Euler scheme or other variant as discussed in the previous subsection. 

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