Numerical Discretization and Solution Techniques

As can be seen from the last section, this expression holds for closed- and open-shell systems. If we add the electron-nucleus potential, this is just the potential of the ion, which the electron leaves behind, i.e., —(Z — N + l)/r. 

In this chapter we have seen how all angular dependencies can be integrated out analytically because of the spherical symmetry of the central field potential of an atom. We are now left with the task to determine the yet unknown radial functions, for which we derived the self-consistent field equations based upon the variational minimax principle. We now address the numerical solution of these equations. The mean-field potential in the set of coupled SCF equations is the reason why we cannot solve them analytically. Hence, the radial functions need to be approximated in some way in 

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Computational issues that are pertinent in MD simulations are time complexity of the force calculations and the accuracy of the particle trajectories including other necessary quantitative measures. These two issues overwhelm computational scientists in several ways. MD simulations are done for long time periods and since numerical integration techniques involve discretization errors and stability restrictions which when not put in check, may corrupt the numerical solutions in such a way that they do not have any meaning and therefore, no useful inferences can be drawn from them. Different strategies such as globally stable numerical integrators and multiple time steps implementations have been used in this respect (see [27, 31]). 

Analytical solutions of the self-preserving distribution do exist for some coalescence kernels, and such behavior is sometimes seen in practice (see Fig. 40). For most practical applications, numerical solutions to the population balance are necessary. Several numerical solution techniques have been proposed. It is usual to break the size range into discrete intervals and then solve the series of ordinary differential equations that result. A geometric discretization reduces the number of size intervals (and equations) that are required. Litster, Smit and Hounslow (1995) give a general discretized population balance for nucleation, growth and coalescence. Figure 41 illustrates the evolution of the size distribution for coalescence alone, based on the kernel of Ennis Adetayo (1994). 

Another potential solution technique appropriate for the packed bed reactor model is the method of characteristics. This procedure is suitable for hyperbolic partial differential equations of the form obtained from the energy balance for the gas and catalyst and the mass balances if axial dispersion is neglected and if the radial dimension is first discretized by a technique such as orthogonal collocation. The thermal well energy balance would still require a numerical technique that is not limited to hyperbolic systems since axial conduction in the well is expected to be significant. 

When non-linearities are included in the analysis, we must also solve the domain integral in the integral formulations. Several methods have been developed to approximate this integral. As a matter of fact, at the international conferences on boundary elements, organized every year since 1978 [43], numerous papers on different and novel techniques to approximate the domain integral have been presented in order to make the BEM applicable to complex non-linear and time dependent problems. Many of these papers were pointing out the difficulties of extending the BEM to such applications. The main drawback in most of the techniques was the need to discretize the domain into a series of internal cells to deal with the terms not taken to the boundary by application of the fundamental solution, such as non-linear terms. 

Typically, the numerical solutions techniques used are very specific to the problem. Particularly challenging problems include moving front problems where concentration profiles, for example, may vary widely over a short distance but may not change much at other spatial locations. The spatial discretization must be small close to the front for accuracy and numerical stability, but must be larger at other locations to reduce computation time. Various adaptive grid techniques to change the spatial step sizes have been developed for these problems. One of the more common codes to solve fluid-flow-related problems is FLUENT. 

D time-dependent solution of the Navicr -Stokes equations. The main reason we do not discuss these flows here is that the analytical solution techniques that we develop have had relatively little impact on the analysis or understanding of turbulent flows. The most powerfifl theoretical tools for turbulence research and for the prediction of turbulent flows are currently direct numerical solutions (DNS) of the Navier Stokes equation, typically by use of spectral techniques for discretization. Again, the interested reader will find many texts and references to modem work on turbulent flows.2... 

Before we finish this subsection, we would like to discuss the practical limitations of the Poincare sections, which require Lagrangian particle tracking for extended times. In reality. Fig. 2 presents stroboscopic images of the same four particles passing through thousands of mixing block boundaries. This has two basic implications. First, numerical calculation of the Poincare sections requires either analytical solutions or high-order accurate discretizations of the velocity field. Otherwise, the results may suffer from numerical diffusion and dispersion errors, and the KAM boundaries may not be identified accurately. Second, it is experimentally difficult, if not impossible, to track particles (in three-dimensions) beyond a certain distance allowed by the field of view of the microscopy technique. Despite these... 

The numerical techniques of Chapter 8 can be used for the simultaneous solution of Equation (9.3) and as many versions of Equation (9.1) as are necessary. The methods are unchanged except for the discretization stability criterion and the wall boundary condition. When the velocity profile is flat, the stability criterion is most demanding when at the centerline ... 

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