Other discretisation schemes

as in Sect. 3.1.3 and eliminating derivative terms other than 2 2 

3 C/3X from the four equations, yields the five-point formula 

Similarly, one could attempt to improve the 3C/3T discretisation (other than by using Runge-Kutta integration). In effect, the Crank-Nicolson scheme does this by specifying a central difference approximation at T+J 8T. The same can be done at T by using the Richardson (1911) formula (denoting time steps by the index k)  

Lapidus and Finder (1982) list an interesting assembly of higher-order and asymmetric forms and predictor-corrector methods (see their Sect. 4.6). Of these, the Saul yev (1964) and Liu (1969) forms should be looked into, as they allow relaxation of the X limits while being fairly easy to implement. 

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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). 

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... 

Classification of Simulation Methods by Time Stepping Scheme Commercial flow simulators generally discretise time derivatives using a first order finite difference formula (Euler s method). The time derivative thus involves the difference of functions at the end and at the start of each time step. All other terms in the equations are discretised to involve functions evaluated at the start and the end of each time step. The pressure always appears at the end of the time step and one says that the pressure is implicit. Saturations appear at the end of the time step in the fully implicit approach. The saturation... 

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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