Extrapolation methods in numerical integration

Extrapolation methods have been used for many years for numerical integration. The most well-known of these methods is Romberg integration. A survey by Joyce on the use of extrapolation in numerical analysis appeared in 1971 in which a substantial portion is devoted to numerical integration. In this paper, we shall survey progress made in this field since 1971. The topics surveyed include partition-extrapolation methods for dealing with singular integrands, the work of Lyness and others in generating asymptotic expansions for the error functional in one and several dimensions, the work of de Doncker and others on adaptive extrapolation and the work of Sidi and others on the evaluation of highly oscillatory infinite integrals by extrapolation. Other extrapolation techniques will be mentioned briefly.     

Error expansion for a first-order upwind difference scheme applied to a model convection-diffusion problem

A singularly perturbed convection–diffusion problem isconsidered. The problem is discretized using a simple first-orderupwind difference scheme on general meshes. We derive an expansionof the error of the scheme that enables uniform error boundswith respect to the perturbation parameter in the discrete maximumnorm for both a defect correction method and the Richardsonextrapolation technique. This generalizes and simplifies resultsobtained in earlier publications by Fröhner et al.(2001,Numer. Algorithms, 26, 281–299) and by Natividad &Stynes (2003, Appl. Numer. Math., 45, 315–329). Numericalexperiments complement our theoretical results.     

Richardson extrapolation in boundary value problems for differential equations with nonregular right-hand side

When the finite-difference method is used to solve initial- or boundary value problems with smooth data functions, the accuracy of the numerical results may be considerably improved by acceleration techniques like Richardson extrapolation. However, the success of such a technique is doubtful in cases were the right-hand side or the coefficients of the equation are not sufficiently smooth, because the validity of an asymptotic error expansion — which is the theoretical prerequisite for the convergence analysis of the Richardson extrapolation — is not a priori obvious. In this work we show that the Richardson extrapolation may be successfully applied to the finite-difference solutions of boundary value problems for ordinary second-order linear differential equations with a nonregular right-hand side. We present some numerical results confirming our conclusions.     

An extrapolation procedure to obtain the total fringe number from Gouy fringe pattern data

A rapid new procedure is described for getting the total number of fringes J from Gouy fringe pattern data. This PQ method is exact and the results excellent (within 0.01–0.03 fringe) for ideal systems (_j=0 for all j, Q₀=0). Such systems include most binaries; for these, the diffusion coefficient is either constant or a polynomial function of concentration with small concentration differences. For multicomponent systems and some binaries, Q₀ can be significantly different from 0. In these cases, the PQ method unambiguously gives the integer number of fringes. If in addition Q₀/Q₁ is larger than 2.0, then J obtained from a second extrapolation procedure is also good.     

An application of convergence acceleration techniques to a class of two-point boundary value problems on a semi-infinite domain

For boundary value problems posed on unbounded domains it is often appropriate to impose a boundary condition at infinity. For certain classes of boundary value problem obvious numerical difficulties can be avoided by truncating the unbounded domain and solving a sequence of finite domain problems instead. We introduce a novel technique which is straightforward to implement and which exploits information contained in this sequence in order to extrapolate to the unbounded case. The technique introduces a new and interesting application of a variety of convergence acceleration algorithms.     

An automatic integration procedure for infinite range integrals involving oscillatory kernels

Let the real functionsK(x) andL(x) be such thatM(x)=K(x)+iL(x)=e^ix g(x), whereg(x) is infinitely differentiable for all largex and is non-oscillatory at infinity. We develop an efficient automatic quadrature procedure for numerically computing the integrals _a K(t)f(t) and _a L(t)f(t)dt, where the functionf(t) is smooth and nonoscillatory at infinity. One such example for which we also provide numerical results is that for whichK(x)=J (x) andL(x)=Y (x), whereJ (x) andY (x) are the Bessel functions of order . The procedure involves the use of an automatic scheme for Fourier integrals and the modified W-transformation which is used for computing oscillatory infinite integrals.     

An extrapolation method for a class of boundary integral equations

Boundary value problems of the third kind are converted into boundary integral equations of the second kind with periodic logarithmic kernels by using Green's formulas. For solving the induced boundary integral equations, a Nyström scheme and its extrapolation method are derived for periodic Fredholm integral equations of the second kind with logarithmic singularity. Asymptotic expansions for the approximate solutions obtained by the Nyström scheme are developed to analyze the extrapolation method. Some computational aspects of the methods are considered, and two numerical examples are given to illustrate the acceleration of convergence.

     

Improvement and application of two extrapolation methods to FT-IR data

The applicability of two non-parametric extrapolation methods to FT-IR absorptance spectra is investigated. The first method minimizes those parts of the spectrum which do not satisfy a given constraint, while the second one changes them in each iteration through the true values.     

双外推法研究FeC2O4·2H2O脱水过程的动力学机理

On the basis of the Coats-Redfern's integral equation and Ozawa's equation. the probable mechanism of the dehydration process of FeC2O4 2H2O was investigated using double extrapolation. The dehydration Process includes two steps. The first step is the nuclear producing and growing process(n=1.5), G(α)= [ln(1-α)]1/1.5; the second step is a two-dimensional diffusion process, G(α)=(1-α)ln(1-α)+a: the corresponding kinetic parameters were determined.     

Analysis of extrapolation cascadic multigrid method(EXCMG)

Based on an asymptotic expansion of finite element,a new extrapolation formula and extrapolation cascadic multigrid method(EXCMG)are proposed,in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid.In the case of triple grids,the error of the new initial value is analyzed in detail.A larger scale computation is completed in PC.