Numerical method, description

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. 

Labels are distinguished based on whether they are context dependent or context-free. Context-dependent labels require simultaneous consideration of time records from more than one process variable context-free labels do not. Thus, generating context-free trend, landmark, and fault descriptions is considerably more simple than generating context-dependent descriptions. Context-free situations can take advantage of numerous methods for common, yet useful, interpretations. Context-dependent situations, however, require the application of considerable process knowledge to get a useful interpretation. In these situations, performance is dependent on the availability, coverage, and distribution of labeled process data from... 

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. 

Many interesting phenomena can arise in nonlinear periodic structures that possess the Kerr nonlinearity. For analytic description of such effects, the slowly varying amplitude (or envelope) approximation is usually applied. Alternatively, in order to avoid any approximation, we can use various numerical methods that solve Maxwell s equations or the wave equation directly. Examples of these rigorous methods that were applied to the modelling of nonlinear periodical structures are the finite-difference time-domain method, transmission-line modelling and the finite-element frequency-domain method." ... 

Today, there an established software tool set does exist for the primary task, the calculation of modes and the description of field propagation. Approaches based on the finite element method (FEM) and finite differences (FD) are popular since long and can be applied to complex problems . The wave matching method, Green functions approaches, and many more schemes are used. But, as a matter of fact, the more dominant numerical methods are, the more the user has to scrutinize the results from the physical point of view. Recent mathematical methods, which can control accuracy absolutely - at least if the problem is well posed, help the design engineer with this. ... 

The final move of the Methods section involves the description of statistical, computational, or other mathematical methods used to derive or analyze data. This move is required only if numerical methods were part of the work. Excerpts 3W... 

By now, you likely have a solid draft of your Methods section, including a description of materials, experimental methods, and numerical methods (if applicable) hence, it is time to revise and edit your Methods section as a whole. We recommend that you reread and edit your work, focusing on each of the following areas and using chapter 18 to guide you. 

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,... 

Ibid, pp 527-37 [A brief description of the following numerical methods for calculation a) "Finite Difference Scheme in Lagrangian Coordinates , previously described by Goad (Ref 5) b) Particle-in Cell Method, previously described by Evans Harlow (Ref 1)... 

The results obtained for the stochastic model show that surface reactions are well-suited for a description in terms of the master equations. Since this infinite set of equations cannot be solved analytically, numerical methods must be used for solving it. In previous Sections we have studied the catalytic oxidation of CO over a metal surface with the help of a similar stochastic model. The results are in good agreement with MC and CA simulations. In this Section we have introduced a much more complex system which takes into account the state of catalyst sites and the diffusion of H atoms. Due to this complicated model, MC and in some respect CA simulations cannot be used to study this system in detail because of the tremendous amount of required computer time. However, the stochastic ansatz permits to study very complex systems including the distribution of special surface sites and correlated initial conditions for the surface and the coverages of particles. This model can be easily extended to more realistic models by introducing more aspects of the reaction mechanism. Moreover, other systems can be represented by this ansatz. Therefore, this stochastic model represents an elegant alternative to the simulation of surface reaction systems via MC or CA simulations. 

In many cases ordinary differential equations (ODEs) provide adequate models of chemical reactors. When partial differential equations become necessary, their discretization will again lead to large systems of ODEs. Numerical methods for the location, continuation and stability analysis of periodic and quasi-periodic trajectories of systems of coupled nonlinear ODEs (both autonomous and nonautonomous) are extensively used in this work. We are not concerned with the numerical description of deterministic chaotic trajectories where they occur, we have merely inferred them from bifurcation sequences known to lead to deterministic chaos. Extensive literature, as well as a wide choice of algorithms, is available for the numerical analysis of periodic trajectories (Keller, 1976,1977 Curry, 1979 Doedel, 1981 Seydel, 1981 Schwartz, 1983 Kubicek and Hlavacek, 1983 Aluko and Chang, 1984). 

This chapter continues with a description of a few basic numerical methods and their underlying principles. However, a solid first course in numerical analysis cannot be replaced by the concise intuitive explanations of numerical methods and phenomena that follow below. 

Equations 7.4 and 7.5 form a system of differential equations for which no analytical solution is known. Thus, the description of the behavior of the semi-batch reactor with time requires the use of numerical methods for the integration of the differential equations. Usually, it is convenient to use parameters which are more process-related to describe the material balance. One is the stoichiometric ratio between the two reactants A and B ... 

One can consider equations (3.37), (3.39) and (3.41) to be a basic system of equations for description of dynamics of entangled systems. The system can be investigated analytically in linear approximation as will be demonstrated in the ensuing chapters. However, to study these non-linear equations in complete form, one has to use numerical methods of simulation of the stochastic processes for the particle coordinates. 

Once the method has been established and validated, it should be described in full detail such that it can be carried out by any other analyst. Besides the numerous experimental details relating to the chemicals, solvents and solutions used and the chromatographic parameters, important observations such as for instance the findings about the stability of standard solutions should be laid down appropriately in the method description as notes or remarks. But potential health risks to the analytical operator should also be addressed, for instance in a warning note at the beginning of the method description. The following structure of a method description, which was agreed upon as a CEN standard format, is a recommended example. 

The method has a long history. The name MOL seems to have become estabhshed around 1960. Before this, various authors either used the word line [254] or expressions like on certain lines [330] or a description of the idea. In the book by Kantorovich and Krylov [330], there is a reference to a 1934 paper [329]. It is also cited by Liskovets [366] as a source paper, along with Rothe (1930) [475], who might be the first. Hartree and Womersley [296] use, in their summary, the words approximating by use of finite intervals in one variable, and integrating exactly in the other variable . The book by Schiesser [497] is the standard work now (he calls the method NUMOL, for numerical method of hnes). Electrochemical use of MOL has been sparse. Lemos and coworkers [357, 359, 360] have investigated the method, using... 

Eisenberg, M., Descriptive Simulation Combining Symbolic and Numerical Methods in the Analysis of Chemical Reaction Mechanisms. Technical Report, Laboratory for Computer Science, MIT, Cambridge, Mass., 1989. 

Chapter 4 is devoted to the description of stochastic mathematical modelling and the methods used to solve these models such as analytical, asymptotic or numerical methods. The evolution of processes is then analyzed by using different concepts, theories and methods. The concept of Markov chains or of complete connected chains, probability balance, the similarity between the Fokker-Plank-Kolmogorov equation and the property transport equation, and the stochastic differential equation systems are presented as the basic elements of stochastic process modelling. Mathematical models of the application of continuous and discrete polystochastic processes to chemical engineering processes are discussed. They include liquid and gas flow in a column with a mobile packed bed, mechanical stirring of a liquid in a tank, solid motion in a liquid fluidized bed, species movement and transfer in a porous media. Deep bed filtration and heat exchanger dynamics are also analyzed. 

There are numerous method validation examples in the literature [9-18]. Each company has their own approach and own set of acceptance criteria for different analytical assays, but these approaches must be within the confines of their line unit QA department and be in accordance with any regulatory provisions. In the next section a description for each of the parameters to be validated (figures of merit) are described in detail and examples are given for each. 

An orthogonal collocation method for elliptic partial differential equations is presented and used to solve the equations resulting from a two-phase two-dimensional description of a packed bed. Comparisons are made between the computational results and experimental results obtained from earlier work. Some qualitative discrimination between rival correlations for the two-phase model parameters is possible on the basis of these comparisons. The validity of the numerical method is shown by applying it to a one-phase packed-bed model for which an analytical solution is available problems arising from a discontinuity in the wall boundary condition and from the semi-infinite domain of the differential operator are discussed. 

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.

The theoretical foundation of this software is provided by principles of continuum mechanics, together with improved numerical methods for the solution of the mathematical equations and by the use of pertinent constitutive equations for the description of the rheological behaviour of molten polymers. 

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