Asymptotic integration method

The second approach is a fractional-step method we call asymptotic timestep-splitting. It is developed by consideration of the specific physics of the problem being solved. Stiffness in the governing equations can be handled "asymptotically" as well as implicitly. The individual terms, including those which lead to the stiff behavior, are solved as independently and accurately as possible. Examples of such methods include the Selected Asymptotic Integration Method (4,5) for kinetics problems and the asymptotic slow flow algorithm for hydrodynamic problems where the sound speed is so fast that the pressure is essentially constant (6, 2). ... 

The Selected Asymptotic Integration Method (5) has been utilized for many years at NRL for the solution of the coupled "stiff" ordinary differential equations associated with reactive flow problems. This program has been optimized for the ASC. 

As another application of this method of asymptotic integration, we shall consider the problem of the Fourier coefficients pfU(P t) i 1 the limit of long times. As mentioned above, we do not wish to give here a detailed proof of the transport equation for pk] p] t) (see, for instance, Ref. 31). The main result of this analysis is, however, very simple in the limit of long times (t —> oo), the correlations are entirely determined by the velocity distribution function p< p t). One has ... 

Near the point where the two streams first meet the chemical reaction rate is small and a self-similar frozen-flow solution for Yp applies. This frozen solution has been used as the first term in a series expansion [62] or as the first approximation in an iterative approach [64]. An integral method also has been developed [62], in which ordinary differential equations are solved for the streamwise evolution of parameters that characterize profile shapes. The problem also is well suited for application of activation-energy asymptotics, as may be seen by analogy with [65]. The boundary-layer approximation fails in the downstream region of flame spreading unless the burning velocity is small compared with u it may also fail near the point where the temperature bulge develops because of the rapid onset of heat release there,... 

The computation of the incomplete Gamma functions is a vital part of evaluating of the ERIs for all integral methods except the Rys-Gauss quadrature. In the evaluation of Fm(T ) two formulae can be used depending on the value of the argument, T. Firstly, there is the asymptotic formula in which the Laplace formula (see equation 13) is utilized. 

To some other experts the meaning of the term ab initio is rather clear cut. Their response is that "ab initio" simply means that all atomic/molecular integrals are computed analytically, without recourse to empirical parametrization. They insist that it does not mean that the method is exact nor that the basis set contraction coefficients were obtained without recourse to parametrization. Yet others point out that even the integrals need not be evaluated exactly for a method to be called ab initio, given that, for instance, Gaussian employs several asymptotic and other cutoffs to approximate integral evaluation. 

REFERENCES de Brujin, N. G. Asymptotic Methods in Analysis, Dover, New York (1981) Folland, G. B., Advanced Calculus, Prentice-Hall, Saddle River, N.J. (2002) Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals, Series, and Troducts, Academic, New York (2000) Kaplan, W., Advanced Calculus, 5th ed., Addison-Wesley, Redwood City, Calif. (2003). 

The long-range asymptotic form of the exchange potential, a constituent of the exchange-correlation potential (56), will be discussed in this and the remaining subsections. Since the known derivations of this form are very complicated and involve an analysis of an integral equation of the OP method (see, e.g. [17], [27],... 

Jones, D. S., and M. Kline, 1958. Asymptotic expansions of multiple integrals and the method of stationary phase, J. Math. Phys., 37, 1-28. 

In summary, we have shown that the kinetics of the bimolecular reaction A + B —> 0 with immobile reactants follows equation (6.1.1), even on a fractal lattice, if d is replaced by d, equation (6.1.29). Moreover, the analytical approach based on Kirkwood s superposition approximation [11, 12] may also be applied to fractal lattices and provides the correct asymptotic behaviour of the reactant concentration. Furthermore, an approximative method has been proposed, how to evaluate integrals on fractal lattices, using the polar coordinates of the embedding Euclidean space. 

The inversion of this transform gives a somewhat cumbersome integral, of which the physical meaning is far from obvious, and Lighthill Whitham naturally prefer to elucidate this form the asymptotic behaviour of the transform, by the method of steepest descents. The method presented here also uses the transform without the need for inversion and obtains a description of the wave in terms of its moments. 

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