Algorithms numerical

U. Ascher. Stabilization of invariants of discretized diflferential systems. Numerical Algorithms, 14 1-23, 1997. 

Swarbrick, S. J. and Nassehi, V., 1992a. A new decoupled finite element algorithm for viscoelastic flow. Part 1 numerical algorithm and sample results. Int. J. Numer. Methods Fluids 14, 1367-1376,... 

Computational methods have played an exceedingly important role in understanding the fundamental aspects of shock compression and in solving complex shock-wave problems. Major advances in the numerical algorithms used for solving dynamic problems, coupled with the tremendous increase in computational capabilities, have made many problems tractable that only a few years ago could not have been solved. It is now possible to perform two-dimensional molecular dynamics simulations with a high degree of accuracy, and three-dimensional problems can also be solved with moderate accuracy. 

The numerical solution of the energy balance and momentum balance equations can be combined with flow equations to describe heat transfer and chemical reactions in flow situations. The simulation results can be in various forms numerical, graphical, or pictorial. CFD codes are structured around the numerical algorithms and, to provide easy assess to their solving power, CFD commercial packages incorporate user interfaces to input parameters and observe the results. CFD... 

The commercial CFD codes use the finite volume method, which was originally developed as a special finite difference formulation. The numerical algorithm consists of the following steps ... 

A numerical algorithm for the solution of the system of Eqs. (15), (19) and (51) consists of the expansion of the two-particle functions into a Fourier-Bessel series. We omit all the details of the numerical method they can be found in Refs. 55-58, 85, 86. In Fig. 3 we show a comparison of the total... 

In the numerical solution the matrix structure is evaluated from Eqs. (44)-(46). Then Eqs. (47)-(49) with corresponding closure approximations are solved. Details of the solution have been presented in Refs. 32 and 33. Briefly, the numerical algorithm uses an expansion of the two-particle functions into a Fourier-Bessel series. The three-fold integrations are then reduced to sums of one-dimensional integrations. In the case of hard-sphere potentials, the BGY equation contains the delta function due to the derivative of the pair interactions. Therefore, the integrals in Eqs. (48) and (49) are onefold and contain the contact values of the functions... 

All the early work was concerned with atoms, with Sir William Hartree regarded as the father of the technique. His son, Douglas R. Hartree, published the definitive book, The Calculation of Atomic Structures, in 1957, and in this he derived the atomic HF equations and described numerical algorithms for their solution. Charlotte Froese Fischer was a research student working under the guidance of D. R. Hartree, and she published her own definitive book. The Hartree—Fock Method for Atoms A Numerical Approach in 1977. The Appendix lists a number of freely available atomie structure programs. Most of these can be obtained from the Computer Physics Communications Program Library. 

Maxima, minima and saddle points are stationary points on a potential energy surface characterized by a zero gradient. A (first-order) saddle point is a maximum along just one direction and in general this direction is not known in advance. It must therefore be determined during the course of the optimization. Numerous algorithms have been proposed, and I will finish this chapter by describing a few of the more popular ones. 

There are numerous algorithms of different kinds and quality in routine use for the fast and reliable localization of minima and saddle points on potential energy surfaces (see 47) and refs, therein). Theoretical data about structure and properties of transition states are most interesting due to a lack of experimental facts about activated complexes, whereas there is an abundance of information about educts and products of a reaction. 

The algebraic solution is the classical fitting technique, as exemplified by the linear regression (Chapter 2). The advantage lies in the clear formulation of the numerical algorithm to be used and in the uniqueness of the solution. If one is free to choose the calibration concentrations and the number of... 

Clearly, the extent of exotherm-generated temperature overshoot predicted by the Chiao and finite element models differs substantially. The finite element results were not markedly changed by refining the mesh size or the time increments, so the difference appears to be inherent in the numerical algorithms used. Such comparison is useful in further development of the codes, as it provides a means of pinpointing those model parameters or algorithms which underlie the numerical predictions. These points will be explored more fully in future work. 

Gear, C. W. In Numerical Algorithms Group Library Manual 1984, Routine D02EBF. 

N4SID = numerical algorithms for subspace state space system identification t = time [sec]... 

Dynamic simulation with discrete-time events and constraints. In an effort to go beyond the integer (logical) states of process variables and include quantitative descriptions of temporal profiles of process variables one must develop robust numerical algorithms for the simulation of dynamic systems in the presence of discrete-time events. Research in this area is presently in full bloom and the results would significantly expand the capabilities of the approaches, discussed in this chapter. 

Let us now see how the theory of the wavelet-based decomposition and reconstruction of discrete-time functions can be converted into an efficient numerical algorithm for the multiscale analysis of signals. From Eq. (6b) it is easy to see that, given a discrete-time signal, FqU) we have... 

Numerical calculations were carried out in order to test the whole numerical algorithm and accuracy of the rate calculation. The potential system employed is a nonlinearly transformed model of the separable case [31]. That is... 

The NAG Fortran Library Manual, Mark 16. The numerical algorithms group limited, Oxford, 1993. 

DPMs offer a viable tool to study the macroscopic behavior of assemblies of particles and originate from MD methods. Initiated in the 1950s by Alder and Wainwright (1957), MD is by now a well-developed method with thousands of papers published in the open literature on just the technical and numerical aspects. A thorough discussion of MD techniques can be found in the book by Allen and Tildesley (1990), where the details of both numerical algorithms and computational tricks are presented. Also, Frenkel and Smit (1996) provide a comprehensive introduction to the recipes of classical MD with emphasis on the physics underlying these methods. Nearly all techniques developed for MD can be directly applied to discrete particles models, except the formulation of particle-particle interactions. Based on the mechanism of particle-particle interaction, a granular system may be modeled either as hard-spheres or as soft-spheres. ... 

Finally, we can also mention that laminar-flow systems with non-Newtonian fluids often require special numerical algorithms that are usually not available in CFD codes designed mainly for turbulent flows. 

---