Numerical methods, use

Equation 13-39 is a cubic equation in terms of the larger aspect ratio R2. It can be solved by a numerical method, using the Newton-Raphson method (Appendix D) with a suitable guess value for R2. Alternatively, a trigonometric solution may be used. The algorithm for computing R2 with the trigonometric solution is as follows ... 

General solution of the population balance is complex and normally requires numerical methods. Using the moment transformation of the population balance, however, it is possible to reduce the dimensionality of the population balance to that of the transport equations. It should also be noted, however, that although the mathematical effort to solve the population balance may therefore decrease considerably by use of a moment transformation, it always leads to a loss of information about the distribution of the variables with the particle size or any other internal co-ordinate. Full crystal size distribution (CSD) information can be recovered by numerical inversion of the leading moments (Pope, 1979 Randolph and Larson, 1988), but often just mean values suffice. 

The chemical bonding and the possible existence of non-nuclear maxima (NNM) in the EDDs of simple metals has recently been much debated [13,27-31]. The question of NNM in simple metals is a diverse topic, and the research on the topic has basically addressed three issues. First, what are the topological features of simple metals This question is interesting from a purely mathematical point of view because the number and types of critical points in the EDD have to satisfy the constraints of the crystal symmetry [32], In the case of the hexagonal-close-packed (hep) structure, a critical point network has not yet been theoretically established [28]. The second topic of interest is that if NNM exist in metals what do they mean, and are they important for the physical properties of the material The third and most heavily debated issue is about numerical methods used in the experimental determination of EDDs from Bragg X-ray diffraction data. It is in this respect that the presence of NNM in metals has been intimately tied to the reliability of MEM densities. 

In this section we shall review the numerical methods used for solving Eq. (53). These are chiefly iterative methods. They differ in the procedures according to which the iterates are generated. 

Among the numerous methods used to formulate a mathematical relationship the following are prime examples. 

The numerical method used for solving the heat-transfer equation is similar to that for solving the momentum equation, which is a finite-volume, ALE method (Kashiwa et al., 1994). 

In this section we describe the numerical methods used to study the classical and quantal dynamics of the LiNC/LiCN system. 

The following example illustrates a numerical method used to assess reactor performance when more than one process affects the overall kinetics of the reaction. 

We have seen that Lagrangian PDF methods allow us to express our closures in terms of SDEs for notional particles. Nevertheless, as discussed in detail in Chapter 7, these SDEs must be simulated numerically and are non-linear and coupled to the mean fields through the model coefficients. The numerical methods used to simulate the SDEs are statistical in nature (i.e., Monte-Carlo simulations). The results will thus be subject to statistical error, the magnitude of which depends on the sample size, and deterministic error or bias (Xu and Pope 1999). The purpose of this section is to present a brief introduction to the problem of particle-field estimation. A more detailed description of the statistical error and bias associated with particular simulation codes is presented in Chapter 7. 

Most of the applications of artificial intelligence in chemistry so far have not involved numerical computation as a primary goal. Yet there are aspects of the AI approach to problem-solving which have relevance to computation. In scientific computation, one could view the knowledge base as the set of equations, input variable values, and unit conversions relevant to the problem, and the inference engine the numerical method used to solve the equations. This paper describes such a software system,... 

These models describe the development of surface charge and potential together with ion adsorption in a quantitative manner. They have in common a set of simultaneous equations that can be solved by numerical methods using the appropriate... 

W.B. Goad, LAMS (Los Alamos Scientific Laboratory, Los Alamos, New Mexico) Manuscript 2365(Nov I960) (Description of one of the numerical methods used by A. Vidart et al) (Ref 21) 6) R. Courant K.O. Friedrichs,... 

The following sections will discuss some of the numerous methods used to apply liquid systems to different surfaces. Many of these techniques are employed in both UV and EB curing processes. Some are very specific to UV technology and these will be dealt with separately in connection with that curing method. 

As it will be discussed, while three maxima of the first derivative are observed, the second one is a consequence of the applied numerical method. Using the second derivative values in the last column, local inverse linear interpolation gives V = 3.74 ml and V = 7.13 ml for the two equivalence points. We will see later on how the false end point can be eliminated. 

Detailed analysis leads to the results shown in Fig 6, which compares computations for the open-end and closed-end pipes . Note that dimensionless piston velocities coincide above c/m = 6.7 (r = c/m). The original explanation offered for this effect was that for rc = 6.7 the piston is so light that the rebounding shock cannot overtake it. Subsequent analysis indicated that r < 7.1 the rebounding shock does indeed overtake the piston. This discrepancy was resolved by a more detailed analysis which showed that effects of the rebounding shock of rc < r < 7.1 are so slight as to be indistinguishable by the numerical methods used in the computations... 

W.Y. Yang, W. Cao, T-S. Chung, J Morris, Applied Numerical Methods Using MATLAB, Wiley-Interscience, 2005, 528 pages (ISBN 0471698334)... 

This charge will influence or even prevent the emission of secondary ions. To overcome this charging of insulators there have been numerous methods used, as seen in Table VIII. 

The numerical method used to decompose the chemical shift spectra are detailed in a paper submitted to Macromolecules. 

One of the limitations of dimensional similitude is that it shows no direct quantitative information on the detailed mechanisms of the various rate processes. Employing the basic laws of physical and chemical rate processes to mathematically describe the operation of the system can avert this shortcoming. The resulting mathematical model consists of a set of differential equations that are too complex to solve by analytical methods. Instead, numerical methods using a computerized simulation model can readily be used to obtain a solution of the mathematical model. 

Despite the large number of analytical solutions available for the diffusion equation, their usefulness is restricted to simple geometries and constant diffusion coefficients. However, there are many cases of practical interest where the simplifying assumptions introduced when deriving analytical solutions are unacceptable. Such a case, for example, is the diffusion in polymer systems characterized by concentration-dependent diffusion coefficients.This chapter gives an overview of the most powerful numerical methods used at present for solutions of the diffusion equation. Indeed the application of these methods in practice needs the use of adequate computer programs (software). 

A number of computer programs are discussed in this book. These are all based on relatively simple finite-difference procedures that are developed in the book. While the numerical methods used are relatively simple, it is believed that if the students gain a good understanding of these methods and are exposed to the power of even simple numerical solution procedures, they will have little difficulty in understanding and using more advanced numerical methods. Examples of the use of the computer programs are included in the text. 

For large values of 112 (highly excited states of the r -equation) one can in general obtain more accurate values of E by the phase-integral method than by the numerical method used by Luc-Koenig and Bachelier (1980a), but for small values of ri2 the numerical method is in general more accurate. 

Instabilities are present in the physical problem studied but they are also present in the numerical methods used to simulate these mechanisms. Most high-fidelity numerical schemes required for Computational Fluid D5mamics exhibit low dissipation and therefore multiple non-physical instabilities (wiggles) arise which can require significant efforts to be kept under control [374 362 340]. 

Ignition sequences can also be simulated with the same LES tool. The numerical method used to mimic an ignition by spark plug in the combustion chamber is the addition of the source term uigpark in Eq. (10.16). This source term, defined by Eq. (10.16), is a Gaussian function located at (xq, 2/0, o) near the upper wall between both primary jets and deposited at time f = fo = 0. The spark duration, typical of industrial spark plugs, is... 

Horstmann and Chase [35] have used the mass transfer parameters determined in stirred tank experiments to simulate the breakthrough curves of affinity chromatography experiments. Numerical methods using different computer packages were carried out to solve the differential equations of the stirred tank adsorption and to predict the performances of a packed bed chromatographic column. 

In order to construct a TTT cure diagram, a series of isothermal cures are performed at different temperatures, and the relative rigidity and logarithmic decrement are recorded vs. time. From each damped wave 1/P and A are extracted the numerical methods used have been described... 

Table II gives a general description of the program features such as total number of elements, aqueous species, gases, organic species, redox species, solid species, pressure and temperature ranges over which calculations can be made, an indication of the types of equations used for computing activity coefficients, numerical method used for calculating distribution of species and the total number of iterations required by these models for each of the two test cases. The chemical analyses for the two test cases are summarized in Table III. The seawater compilation was prepared in several units to assure consistency between concentrations for proper entry into the aqueous models.

Stabilizers showing limited solubility in water or a low vapor pressure in air constitute a major problem for the modeling (5, P). The migration is then controlled by the boundary conditions and the numerical method used may not work properly. In these cases, data obtained from pressure testing with stagnant water may not represent the real-life situation with internal flowing water. 

Infrared dichroism is one of numerous methods used to characterize molecular orientation. The degree of anisotropy of the strained pol3rmers may also be accurately characterized by other techniques such as X-ray diffraction, birefringence, sonic modulus, polarized fluorescence and polarized Raman spectroscopy [2]. These techniques directly probe the orientational behavior of macromolecular chains at a molecular level, in contrast to the macroscopic information provided by mechanical measurements. 

The notation is that of Eqs. (2)-(4). Thus, from the usual steady-state experiment we measure r we must use the tracing experiment to find r+ and r. For the conditions used 107), the inlet gas is more than 95% He so that to a good approximation qi = q and qlV can be set equal to 1/t however, with the numerical methods used 107) it is not necessary to make any simplifications. The H2 balance can also be ignored. We now apply the differential equations (Eqs. 2-4) to each pool, with the foUowuig results for balances on the marked molecules ... 

In the case of benzene... Ar clusters a 2.5 fs time step was used very similar results were, however, obtained with shorter time steps of 0.5 or 1.0 fs. In the case of NA base pairs a 1 fs time step was used. The total energy of the cluster was conserved within 5 cm"l during the MD run and this fluctuation originates from the numerical method used. 

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