An asymptotic expansion for product integration applied to Cauchy principal value integrals

Product integration methods for Cauchy principal value integrals based on piecewise Lagrangian interpolation are studied. It is shown that for this class of quadrature methods the truncation error has an asymptotic expansion in integer powers of the step-size, and that a method with an asymptotic expansion in even powers of the step-size does not exist. The relative merits of a quadrature method which employs values of both the integrand and its first derivative and for which the truncation error has an asymptotic expansion in even powers of the step-size are discussed

Mixed method for the product integral on the infinite interval

In this note, quadrature formula is constructed for product integral on the infinite interval I(f) = ∫ w(x)f(x)dx, where w(x) is a weight function and f(x) is a smooth decaying function for x > N (large enough) and piecewise discontinuous function of the first kind on the interval a ≤ x ≤ N. For the approximate method we have reduced infinite interval x [a, ∞) into the interval t[0,1] and used the mixed method: Cubic Newton’s divided difference formula on [0, t3) and Romberg method on [t3,1] with equal step size, ti = t0+ih,i=0, …,n, h=1/n, where t0 = 0,tn=1. Error term is obtained for mixed method on different classes of functions. Finally, numerical examples are presented to validate the method presented

Quadrature algorithms to the luminosity distance with a time-dependent dark energy model

In our previous work, we have proposed two methods for computing the luminosity distance d_{L}^{\Lambda} in LCDM model. In this paper, two effective quadrature algorithms, known as Romberg Integration and composite Gaussian Quadrature, are presented to calculate the luminosity distance d_{L}^{CPL} in the Chevallier-Polarski-Linder parametrization(CPL) model. By comparing the efficiency and accuracy of the two algorithms, we find that the second is more promising. Moreover, we develop another strategy adapted for approximating d_{L}^{\Lambda} in flat LCDM universe. To some extent, our methods can make contributions to the recent numerical stimulation for the investigation of dark energy cosmology.Comment: 12 pages, 3 figures, 3 tables, version accepted for publication in JCAP (http://iopscience.iop.org/1475-7516/2011/11/047

Some quadrature methods for general and singular integrals in one and two dimensions

In this thesis numerical integration in one and two dimensions is considered. In chapter two transformation methods are considered primarily for singular integrals and methods of computing the transformations themselves are derived. The well-known transformation based on the IMT rule and error function are extended to non-standard functions. The implementation of these rules and their performances are demonstrated. These transformations are then extended to two-dimensions and are used to develop accurate rules for integrating singular integrals. In addition to this, a polynomial transformation with the aim of the reduction in the number of function evaluations is also considered and the resultant product rule is applied to two-dimensional non-singular integrals. Finally, the use of monomials in the construction of integration rules for non-singular two-dimensional integrals is considered and some rules developed. In all these situations the rules developed are tested and compared with existing methods. The results show that the new rules compare favourably with existing ones

Studies in numerical quadrature

Various types of quadrature formulae for oscillatory integrals are studied with a view to improving the accuracy of existing techniques. Concentration is directed towards the production of practical algorithms which facilitate the efficient evaluation of integrals of this type arising in applications. [Continues.

Spectral Energy Distributions of Passive T Tauri Disks: Inclination

We compute spectral energy distributions (SEDs) for passive T Tauri disks viewed at arbitrary inclinations. Semi-analytic models of disks in radiative and hydrostatic equilibrium are employed. Over viewing angles for which the flared disk does not occult the central star, the SED varies negligibly with inclination. For such aspects, the SED shortward of ~80 microns is particularly insensitive to orientation, since short wavelength disk emission is dominated by superheated surface layers which are optically thin. The SED of a nearly edge-on disk is that of a class I source. The outer disk occults inner disk regions, and emission shortward of ~30 microns is dramatically extinguished. Spectral features from dust grains may appear in absorption. However, millimeter wavelength fluxes decrease by at most a factor of 2 from face-on to edge-on orientations. We present illustrative applications of our SED models. The class I source 04108+2803B is considered a T Tauri star hidden from view by an inclined circumstellar disk. Fits to its observed SED yield model-dependent values for the disk mass of ~0.015 solar masses and a disk inclination of ~65 degrees relative to face-on. The class II source GM Aur represents a T Tauri star unobscured by its circumstellar disk. Fitted parameters include a disk mass of \~0.050 solar masses and an inclination of ~60 degrees.Comment: Accepted to ApJ, 20 pages, 7 figures, aaspp4.st

Efficient and Accurate Computation of Non-Negative Anisotropic Group Scattering Cross Sections for Discrete Ordinates and Monte Carlo Radiation Transport

A new method for approximating anisotropic, multi-group scatter cross sections for use in discretized and Monte Carlo multi-group neutron transport is presented. The new method eliminates unphysical artifacts such as negative group scatter cross sections and falsely positive cross sections. Additionally, when combined with the discrete elements angular quadrature method, the new cross sections eliminate the lack of angular support in the discrete ordinates quadrature method. The new method generates piecewise-average group-to-group scatter cross sections. The accuracy and efficiency for calculating the discrete elements cross sections has improved by many orders of magnitude compared to DelGrande and Mathews previous implementation. The new cross sections have extended the discrete elements method to all neutron-producing representations in the Evaluated Nuclear Data Files. The new cross section method has been validated and tested with the cross section generation code, NJOY. Results of transport calculations using discrete elements, discrete ordinates, and Monte Carlo methods for two, one-dimensional slab geometry problems are compared