An adaptive Gaussian quadrature for the Voigt function

We evaluate an adaptive Gaussian quadrature integration scheme suitable for the numerical evaluation of generalized redistribution in frequency functions. The latter are indispensable ingredients for “full non-LTE” radiation transfer computations, assuming potential deviations of the velocity distribution of massive particles from the usual Maxwell–Boltzmann distribution. A first validation is made with computations of the usual Voigt profile.

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An Algorithm for the Numerical Evaluation of Certain Finite Part Integrals

ISRN Applied Mathematics ◽

10.5402/2011/571583 ◽

2011 ◽

Vol 2011 ◽

pp. 1-21

Author(s):

Samir A. Ashour ◽

Hany M. Ahmed

Keyword(s):

Numerical Evaluation ◽

Gaussian Quadrature ◽

Quadrature Rules ◽

Finite Part ◽

Cauchy Principal ◽

Cauchy Principal Value ◽

Cauchy Principal Value Integrals ◽

Principal Value

Many algorithms that have been proposed for the numerical evaluation of Cauchy principal value integrals are numerically unstable. In this work we present some formulae to evaluate the known Gaussian quadrature rules for finite part integrals , and extend Clenshow's algorithm to evaluate these integrals in a stable way.

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Numerical evaluation of Coulomb integrals for 1, 2 and 3-electron distance operators, RC1-Nrd1-M, RC1-Nr12-M and R12-Nr13-M with real (N, M) and the Descartes product of 3 dimension common density functional numerical integration scheme

10.1063/1.5114496 ◽

2019 ◽

Author(s):

Sandor Kristyan

Keyword(s):

Numerical Integration ◽

Density Functional ◽

Numerical Evaluation ◽

Integration Scheme ◽

Numerical Integration Scheme ◽

Coulomb Integrals

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Adaptive Gaussian quadrature over surface splines in high‐voltage field computation

COMPEL The International Journal for Computation and Mathematics in Electrical and Electronic Engineering ◽

10.1108/03321640010334622 ◽

2000 ◽

Vol 19 (3) ◽

pp. 829-849

Author(s):

C. Vetter ◽

H. Singer

Keyword(s):

High Voltage ◽

Gaussian Quadrature ◽

Field Computation ◽

Surface Splines ◽

Adaptive Gaussian Quadrature

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Laser Investigations of the Dynamics of Molecule-Surface Interaction

MRS Proceedings ◽

10.1557/proc-29-243 ◽

1983 ◽

Vol 29 ◽

Cited By ~ 3

Author(s):

J. Hager ◽

H. Walther

Keyword(s):

Energy Distribution ◽

Surface Temperature ◽

Velocity Distribution ◽

Average Velocity ◽

Boltzmann Distribution ◽

Rotational Energy ◽

Solid Surfaces ◽

Measured Velocity ◽

Small Influence ◽

Molecule Surface

ABSTRACTThe internal energy distribution of NO molecules scattered from different solid surfaces (Pt(111), graphite, and Pt(111) covered with various adlayers) was investigated by the laser-induced fluorescence method. In the case of the NO/graphite system, moreover, the velocity distribution of the scattered molecules could be measured in a time-offlight experiment. The rotational energy distribution, which can always be described as a Boltzmann distribution, exhibits only partial accommodation to the surface temperature for all surfaces investigated. The measurements of the velocity of the NO molecules scattered from the graphite surface show only a small influence of the surface temperature on the average velocity and on the velocity distribution. Furthermore, the measured velocity distribution is independent of the final rotational state of the scattered molecules. On the basis of these results, a rather complete description of the behavior of the NO molecules during the scattering process can be presented.

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A Modified Gaussian Integration Scheme in Element Free Method

Journal of Mechanics ◽

10.1017/s1727719100001982 ◽

2002 ◽

Vol 18 (1) ◽

pp. 17-27

Author(s):

Jopan Sheng ◽

Chung-Yue Wang ◽

Kuo-Jui Shen

Keyword(s):

Numerical Integration ◽

Solid Mechanics ◽

Gaussian Quadrature ◽

Quadrature Rule ◽

Integration Scheme ◽

Quadrature Point ◽

Physical Domain ◽

Gaussian Integration ◽

Numerical Integration Scheme ◽

Element Free

ABSTRACTIn this paper, a modified numerical integration scheme is presented that improves the accuracy of the numerical integration of the Galerkin weak form, within the integration cells of the analyzed domain in the element-free methods. A geometrical interpretation of the Gaussian quadrature rule is introduced to map the effective weighting territory of each quadrature point in an integration cell. Then, the conventional quadrature rule is extended to cover the overlapping area between the weighting territory of each quadrature point and the physical domain. This modified numerical integration scheme can lessen the errors due to misalignment between the integration cell and the boundary or interface of the physical domain. Some numerical examples illustrate that this newly proposed integration scheme for element-free methods does effectively improve the accuracy when solving solid mechanics problems.

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