Discretisation time derivative

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... 

The time derivative is still a central difference but the spatial second derivative now leaves out the central point, substituting for it the mean of the past and future points. Thus, the discretisation is... 

The above set of odes is now solved, choosing some algorithm. Nothing has been specified about the homogeneous chemical reaction function F(C), but it will add terms to the matrix W when specified. After the time derivative is discretised in some way, the equation can be rearranged into the same form as described in Chap. 8 and solved using the same methods or, as mentioned above, solved using a professional ode or DAE solver. 

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. 

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... 

MOL is intimately bound up with another method, that of using DAE sets. It can be thought of as an extension of MOL. Therefore, MOL should be described here, and it is in fact simple. In the most popular form of MOL, the diffusion equation is discretised on a grid in the spatial dimension(s) only, leaving the time derivative as it is. This results in a set of ordinary differential equations (9.53) as seen, for... 

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. 

Firstly, the discretisation itself is described. We restrict the discussion to the BI time integration, in order to focus on the spatial discretisations. The program UMDE DIRECT in fact uses BI as the first step, then three-point BDF, which produces second-order accuracy with respect to ST, this being the rational BDF startup described in Chap. 4, page 59. Take a point away from the boundaries, indices i (for Z) and j (for R). The discretisation at concentration (r j of the pde (12.17) has three derivative terms, all to be discretised using four-point formulas. The coefficients can be precalculated. For the row along Z, there are, for each 0 < Z < Zmox, that is, 0 < i < n-z, four coefficients for the approximations... 

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. 

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... 

The derivative is formed in the outward normal direction, and a and can be dependent on time. For the discretisation of (2.253) it is most convenient if the boundary coincides with a grid line, Fig. 2.45, as the boundary temperature which appears in (2.253) can immediately be used in the difference formula. The replacement of the derivative d d/dn by the central difference quotient requires grid points outside the body, namely the temperatures i9k or ()k,, which, in conjunction with the boundary condition, can be eliminated from the difference equations. 

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... 

The three points thus give us a new point at the next time. Compare this with the box-method expression, Eq. 3.5 they are identical. In fact, one might say that the box-method derives Fick s second diffusion equation in discrete form. Although Eqs. 3.5 and 3.26 are identical, it is clear that the point method derivation is much faster with only a little practice, the discretisation formulae 3.10 to 3.13 are easily memorised and expressions like Eq. 3.26 can be written down straight from the diffusion equation. Furthermore, it will be seen that if the... 

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... 

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. 

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