Numerical computational methods

D. P. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Computation Numerical Methods, Prentice-Hall, Englewood Cliifs, NJ, 1989. 

Since the coming of the computers, numerical methods have found an impressive development. 

The offered method by calculation of roles for species and individual steps of complex chemical reactions with consideration for the selected target characteristics is sufficiently comprehensive and harmonized with computing numerical methods. And we think that the information obtained is illustrative and physically and chemically easy to interpret. 

Computative numerical methods, automation technology, and artificial intelligence are combined in the software to carry out HPLC method development completely automatically. 

In this section we present several numerical teclmiques that are conmronly used to solve the Sclirodinger equation for scattering processes. Because the potential energy fiinctions used in many chemical physics problems are complicated (but known to reasonable precision), new numerical methods have played an important role in extending the domain of application of scattering theory. Indeed, although much of the fomial development of the previous sections was known 30 years ago, the numerical methods (and computers) needed to put this fomialism to work have only been developed since then. 

Okunbor, D.I., Skeel, R.D. Canonical numerical methods for molecular dynamics simulations. J. Comput. Chem. 15 (1994) 72-79. 

Fig. 3. Quantum solution of the test system of 3.3 for e = 1/100. computed numerically using Fourier pseudospectral methods in space and a syraplectic discretization in time. Reduced g -density f t)j dg versus t and qF Initial...

How can Equation (11.79) be solved Before computers were available only simple ihapes could be considered. For example, proteins were modelled as spheres or ellipses Tanford-Kirkwood theory) DNA as a uniformly charged cylinder and membranes as planes (Gouy-Chapman theory). With computers, numerical approaches can be used to solve the Poisson-Boltzmann equation. A variety of numerical methods can be employed, including finite element and boundary element methods, but we will restrict our discussion to the finite difference method first introduced for proteins by Warwicker and Watson [Warwicker and Watson 1982]. Several groups have implemented this method here we concentrate on the work of Honig s group, whose DelPhi program has been widely used. 

Three basic approaches have been used to solve the equations of motion. For relatively simple configurations, direct solution is possible. For complex configurations, numerical methods can be employed. For many practical situations, particularly three-dimensional or one-of-a-kind configurations, scale modeling is employed and the results are interpreted in terms of dimensionless groups. This section outlines the procedures employed and the limitations of these approaches (see Computer-aided engineering (CAE)). 

This equation relates the (instantaneous) copolymer composition with the monomer feed of M and M2. Values for and are usually determined by graphical methods (9,10). Today, with the prevalence of powerful desktop computers, numerical minimisa tion methods are often used (11—14). 

Much professional software is devoted to this problem. A diskette for sets of differential and algebraic equations with parameters to be found by this method is by Constantinides Applied Numerical Methods with Personal Computers, McGraw-Hill, 1987). 

The development of mathemafical models is described in several of the general references [Giiiochon et al., Rhee et al., Riithven, Riithven et al., Suzuki, Tien, Wankat, and Yang]. See also Finlayson [Numerical Methods for Problems with Moving Front.s, Ravenna Park, Washington, 1992 Holland and Liapis, Computer Methods for Solving Dynamic Separation Problems, McGraw-Hill, New York, 1982 Villadsen and Michelsen, Solution of Differential Equation Models by... 

Although much as been done, much work remains. Improved material models for anisotropic materials, brittle materials, and chemically reacting materials challenge the numerical methods to provide greater accuracy and challenge the computer manufacturers to provide more memory and speed. Phenomena with different time and length scales need to be coupled so shock waves, structural motions, electromagnetic, and thermal effects can be analyzed in a consistent manner. Smarter codes must be developed to adapt the mesh and solution techniques to optimize the accuracy without human intervention. 

Fiber and Karplus [38] presented an effective set of numerical methods for computing the reaction paths based on this approximation. First the path is discretized—it is expressed as a chain of intermediate configurations of the system rj,. The line integrals of Fq. (19) are then written as... 

Norris, A. C., Computational Chemistry—An Introduction to Numerical Methods, John Wiley Sons Ltd., 1981. 

Equations 5-110, 5-112, 5-113, and 5-114 are first order differential equations and the Runge-Kutta fourth order numerical method is used to determine the concentrations of A, B, C, and D, with time, with a time increment h = At = 0.5 min for a period of 10 minutes. The computer program BATCH57 determines the concentration profiles at an interval of 0.5 min for 10 minutes. Table 5-6 gives the results of the computer program and Figure 5-16 shows the concentration profiles of A, B, C, and D from the start of the batch reaction to the final time of 10 minutes. 

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 ... 

Additionally, solutions to problems are presented in the text and the accompanying CD contains computer programs (Microsoft Excel spreadsheet and software) for solving modeling problems using numerical methods. The CD also contains colored snapshots on computational fluid mixing in a reactor. Additionally, the CD contains the appendices and conversion table software. 

I hese equations cannot be used directly, and numerical methods are needed to compute the velocity components. The velocity components can be found by implicit differentiation and using an iterative technique.-" ... 

Flynn et al." applied a finite element based numerical model to solve the problem of a push-pull flow with cross-drafts and demonstrate that the results show good agreement with experimental data. They note, however, that the numerical method is time consuming and therefore computationally expensive. 

The standard requires the supplier to have the appropriate resources and equipment (when specified in the contract) to utilize computer-aided product design, engineering, and analysis that is compatible with the customer s and subcontractor systems. It is also required that the supplier be able to use numerical design and drawing data, by computer-aided methods for the manufacture of production tooling and prototypes. 

The combustion-flow interactions should be central in the computation of combustion-generated flow fields. This interaction is fundamentally multidimensional, and can only be computed by the most sophisticated numerical methods. A simpler approach is only possible if the concept of a gas explosion is drastically simplified. The consequence is that the fundamental mechanism of blast generation, the combustion-flow interaction, cannot be modeled with the simplified approach. In this case flame propagation must be formalized as a heat-addition zone that propagates at some prescribed speed. 

In general, discontinuities constitute a problem for numerical methods. Numerical simulation of a blast flow field by conventional, finite-difference schemes results in a solution that becomes increasingly inaccurate. To overcome such problems and to achieve a proper description of gas dynamic discontinuities, extra computational effort is required. Two approaches to this problem are found in the literature on vapor cloud explosions. These approaches differ mainly in the way in which the extra computational effort is spent. 

In the earliest applications of numerical methods for the computation of blast waves, the burst of a pressurized sphere was computed. As the sphere s diameter is reduced and its initial pressure increased, the problem more closely approaches a point-source explosion problem. Brode (1955,1959) used the Lagrangean artificial-viscosity approach, which was the state of the art of that time. He analyzed blasts produced by both aforementioned sources. The decaying blast wave was simulated, and blast wave properties were registered as a function of distance. The code reproduced experimentally observed phenomena, such as overexpansion, subsequent recompression, and the formation of a secondary wave. It was found that the shape of the blast wave at some distance was independent of source properties. 

Appendix F is a case study by Hjertager et al. illustrating the above method. Such numerical methods will become more widely used in the long term. These techniques will probably remain research tools, rather than routine evaluation methods, until such time as available computing power and algorithm efficiency greatly increase. 

Numerical optimizations are available for methods lacking analytic gradients (first derivatives of the energy), but they are much, much slower. Similarly, frequencies may be computed numerically for methods without analytic second derivatives. 

With a computer program, which solved Eqs. (5) and (7) for We by numerical method, Weber numbers were obtained from layer to layer for each injection molding operation. The calculated points were then connected to curves, depending on the normalized thickness. Figure 17 shows these theoretically deduced Weix)... 

The numerical approaches to the solution of the Laplace equation usually demand access to minicomputers with fast processing capabilities. Numerical methods of this sort are essential when the electrolyte is unconfined, as for an off-shore rig or a submarine hull. However, where the electrolyte is confined, as within essentially cylindrical equipment such as pipework and heat-exchangers, or for restricted electrolyte depths, a simpler modelling procedure may be adopted in the case of electrolytes of good conductivity, such as sea-water . This simpler procedure enables computation to be carried out on small, desk-top microcomputers. 

James, M.L., Smith, G.M. and Wolford, J.C. Applied Numerical Methods for Digital Computation . Harper Row, New York, 1977. 

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