Digital computer problem solving

In this section, the conceptual framework of molecular orbital theory is developed. Applications are presented and problems are given and solved within qualitative and semi-empirical models of electronic structure. Ab Initio approaches to these same matters, whose solutions require the use of digital computers, are treated later in Section 6. Semi-empirical methods, most of which also require access to a computer, are treated in this section and in Appendix F. 

Availability of large digital computers has made possible rigorous solutions of equilibrium-stage models for multicomponent, multistage distillation-type columns to an exactness limited only by the accuracy of the phase equilibrium and enthalpy data utilized. Time and cost requirements for obtaining such solutions are very low compared with the cost of manual solutions. Methods are available that can accurately solve almost any type of distillation-type problem quickly and efficiently. The material presented here covers, in some... 

There are few analytic solutions to the governing equations for interesting problems. The conservation equations are typically solved approximately on digital computers. It is assumed that the sound speeds are real and the system... 

Let us return to our discussion of the prediction of ignition time by thermal conduction models. The problem reduces to the prediction of a heat conduction problem for which many have been analytically solved (e.g. see Reference [13]). Therefore, we will not dwell on these multitudinous solutions, especially since more can be generated by finite difference analysis using digital computers and available software. Instead, we will illustrate the basic theory to relatively simple problems to show the exact nature of their solution and its applicability to data. 

Under the heading "General Case , Ma ek states (p 47) that in order to solve eq (1) with out approximations subject to specific boundary conditions, one has to resort to numerical procedures. G. B. Cook (Refs 6a 7a) treated two problems by means of calcns with. digital computers. First is the case of a slab of solid expl,one face of which was in contact with. a constant-temp bath. In the 2nd case the expl was subjected to a time-dependent heat pulse. In both. cases the time to ignition and the critical condition for ignition are given as... 

F. A. Brusentsev, Mathematical Methods of Solving Some Problems in Solid-State Physics and Structural Chemistry with Digital Computers, Nauk. Dumka, Kiev,... 

In the years since the 2nd Edition, much has happened in electrochemical digital simulation. Problems that ten years ago seemed insurmountable have been solved, such as the thin reaction layer formed by very fast homogeneous reactions, or sets of coupled reactions. Two-dimensional simulations are now commonplace, and with the help of unequal intervals, conformal maps and sparse matrix methods, these too can be solved within a reasonable time. Techniques have been developed that make simulation much more efficient, so that accurate results can be achieved in a short computing time. Stable higher-order methods have been adapted to the electrochemical context. 

The development of ab initio methods, which has come to be known as quantum chemistry, is one of the outstanding cumulative intellectual and technical achievements of the past fifty years. Fifty years ago, at the time of the publication of Herzberg s classic book [14], quantum mechanics had been applied to the calculation of the wave functions of the very simplest molecules, but for most systems the problems were considered intractable. At the turn of the millenium we have come a long way, but many difficult problems remain. Progress has been closely related to the development of the digital computer, and the technical achievements in the use of computers to solve problems in quantum mechanics have been impressive. There has always, however, been an accompanying and continual need for intellectual advances to make use of the technology. This section describes the nature of the problems to be solved, and some of the methods which have been developed to tackle them. 

It should be obvious to the reader by now that finite-difference techniques may be applied to almost any situation with just a little patience and care. Very complicated problems then become quite easy to solve when large digital computer facilities are available. Computer programs for several heat-transfer problems of interest are given in Refs. 17, 19, and 21 of Chap. 3. Finite-element methods for use in conduction heat-transfer problems are discussed in Refs. 9 to 13. A number of software packages are available commercially for use with microcomputers. 

The time method of lines (continuous-space discrete-time) technique is a hybrid computer method for solving partial differential equations. However, in its standard form, the method gives poor results when calculating transient responses for hyperbolic equations. Modifications to the technique, such as the method of decomposition (12), the method of directional differences (13), and the method of characteristics (14) have been used to correct this problem on a hybrid computer. To make a comparison with the distance method of lines and the method of characteristics results, the technique was used by us in its standard form on a digital computer. 

The presentation of this case has been dominated by the graphical construction that is possible, for it is felt that this provides a rapid solution of sufficient accuracy for preliminary design discussions. The actual details of the calculation are of course most expeditiously performed on a digital computer. Here the loci, F, would be stored as tables and used as such and for any given values of X and fx the whole process would be a matter of minutes. We shall now show how the availability of solutions of this simple model allows us to solve other significant design problems. 

Under any conditions Eq. (6) can be solved by simultaneously integrating the four equations (8) and (10) and choosing w at each step to be the value that maximizes the right-hand side of (6). If little is known about the structure of the policy a direct numerical search for this value of w is always safe, and is no great problem for the digital computer. 

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