Discretisation implicit method

We might wish to solve it using an implicit method, for example, BI (Sect. 4.6). Discretising (4.61) then gives... 

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]. 

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... 

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... 

A particularly accurate implicit difference method, which is always stable, has been presented by J. Crank and P. Nicolson [2.65]. In this method the temperatures at the time levels tk and tk+l are used. However the differential equation (2.236) is discretised for the time lying between these two levels tk + At/2. This makes it possible to approximate the derivative (dt>/dt)k+1 2 by means of the accurate central difference quotient... 

To transfer the Crank-Nicolson [2.65] implicit difference method, which is always stable, over to cylindrical coordinates requires the discretisation of the equation... 

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]. 

The Galerkin weighted residual method is employed to formulate the finite element discretisation. An implicit mid-interval backward difference algorithm is implemented to achieve temporal discretisation. With appropriate initial and boundary conditions the set of non-linear coupled governing differential equations can be solved. 

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... 

Whether the simulation is on a direct discretisation of the equations in cylindrical or transformed coordinates, the discretisation process results in a (usually) linear system of ordinary differential equations, that must be solved. In two dimensions, the number of these will often be large and the equation system is banded. One approach is to ignore the sparse nature of the system and simply to solve it, using lower-upper decomposition (LUD) [212]. The method is very simple to apply and has been used [133,213,214]—it is especially appropriate in curvilinear coordinates and multipoint derivative approximations, where the system is of minimal size [214], and can outperform the more obvious method, using a sparse solver such as MA2 8 (see later). However, many simulators tend to prefer other methods, that avoid using implicit solution in two dimensions simultaneously but still are implicit. Of these, two stand out. 

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. 

---