Quadrature discretization method

When Bernie Shizgal arrived at UBC in 1970, his research interests were in applications of kinetic theory to nonequilibrium effects in reactive systems. He subsequently applied kinetic theory methods to the study of electron relaxation in atomic and molecular moderators,46 hot atom chemistry, nucleation,47 rarefied gas dynamics,48 gaseous electronics, and other physical systems. An important area of research has been the kinetic theory description of the high altitude portion of planetary atmospheres, and the escape of atmospheric species.49 An outgrowth of these kinetic theory applications was the development of a spectral method for the solution of differential and integral equations referred to as the quadrature discretization method (QDM), which has been used with considerable success in statistical, quantum, and fluid dynamics.50... 

B. D. Shizgal and H. Chen,/. Chem. Phys., 107,8051 (1997). The Quadrature Discretization Method (QDM) in the Solution of the Fokker-Planck Equation with Nonclassical Basis Functions. 

This part of the chapter is devoted to a few of the popular numerical discretization schemes used to solve the population balance equation for the (fluid) particle size distribution. In this section we discuss the method of moments, the quadrature method of moments (QMOM), the direct quadrature method of moments (DQMOM), the discrete method, the chzss method, the multi-group method, and the least squares method. 

Bayesian probability theory157 can also be applied to the problem of NMR parameter estimation this approach incorporates prior knowledge of the NMR parameters and is particularly useful at short aquisition times158 and when the FID contains few data points.159 Bayesian analysis gives more precise estimates of the NMR parameters than do methods based on the discrete Fourier transform (DFT).160 The amplitudes can be estimated independently of the phase, frequency and decay constants of the resonances.161 For the usual method of quadrature detection, it is appropriate to apply this technique to the two quadrature signals in the time domain.162-164... 

Formulated in this way, the problem reduces to one originally considered by Gauss (see, e.g., Stroud and Secrest, 1966). Shibata et al. (1987) review and extend methods for doing that. The important point is that theorems are available that can be used to determine the best quadrature points and the corresponding weighting constants, while in the case of a discrete description the choice of... 

The first step in the regularization method is the discretization of the integral equation by a quadrature. Thus, integral equation (10) needs to be transformed into a system of the following linear equations ... 

Orthogonal collocation in the chemical engineering literature refers to the family of collocation methods with discretization grids associated to Gaussian quadrature methods [34, 204]. Spectral collocation methods for partial differential equations with an arbitrary distribution of collocation points are sometimes termed pseudo spectral methods [22]. 

As for the pure diffusion problem, the key step of the finite volume method is the integration of the governing equations over the grid cell volume to yield a set of discretized equations at the nodal point P. By use of the Gauss theorem and the midpoint quadrature formula, we obtain ... 

N. Selfuk and N. Kayakol, Evaluation of Angular Quadrature and Spatial Differencing Schemes for Discrete Ordinates Method in Rectangular Furnaces, in R. D. Skocypec, S. T. Thynell, D. A. Kaminski, A. M. Smith, and T. Tong (eds.), Solution Methods for Radiative Transfer in Participating Media, ASME HTD vol. 325, ASME, New York, 1996. 

The K-matrix method is essentially a configuration interaction (Cl) performed at a fixed energy lying in the eontinuum upon a basis of "unperturbed functions that (at the formal level) ineludes both discrete and continuous subsets. It turns the Schrodinger equation into a system of integral equations for the K-matrix elements, which is then transformed into a linear system by a quadrature upon afinite Z/ basis set. 

Finite-difference methods operating on a grid consisting of equidistant points ( Xi, Xi = ih + Xq) are known to be one of the most accurate techniques available [496]. Additionally, on an equidistant grid all discretized operators appear in a simple form. The uniform step size h allows us to use the Richardson extrapolation method [494,497] for the control of the numerical truncation error. Many methods are available for the discretization of differential equations on equidistant grids and for the integration (quadrature) of functions needed for the calculation of expectation values. 

In order to obtain a system of algebraic equations, the PBE (12.399) and the boundary conditions must be transformed into a discrete form. In spectral methods, the solution function is approximated in terms of a polynomial solution function expansion (12.408). The differentiation of the discrete solution approximation were presented as (12.410) and (12.411). The PBE (12.399) is an integro-differential equation. Thus, appropriate quadrature rules are required for the numerical solution. Integral approximations can be presented on the form ... 

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