Modified moments and Gaussian quadratures

Wheeler, John C.

doi:10.1216/rmj-1974-4-2-287

Cited by 182 publications

(115 citation statements)

References 9 publications

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“…Thus, the matrix L will be diagonal for all the cases considered in Section 6. The multi-dimensional quadrature formula is defined by considering a set of 2n moments in each direction (e.g., R 0 000 ,··· ,R 2n−1 002n−1 ), which are used to compute a set of n univariate weights and abscissas in each spatial direction, using one of the flavors the product-difference (PD) algorithm [27,33,37]. The velocity abscissas R α are then defined through the tensor product of the univariate abscissas in each direction.…”

Section: Quadrature-based Moment Methodsmentioning

confidence: 99%

“…For the remaining moments, Eq. (3.3) provides an optimal approximation in the sense of Gaussian quadrature [27,33,37].…”

Section: Quadrature-based Moment Methodsmentioning

confidence: 99%

“…For third-order quadrature, weights and abscissas in each direction are found by applying the two-node quadrature formulas [27,37] to the three sets of rotated moments with respect to the principal directions:…”

Section: A Boltzmann-enskog Collision Integralmentioning

confidence: 99%

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A Quadrature-Based Kinetic Model for Dilute Non-Isothermal Granular Flows

Passalacqua

Galvin²,

Vedula

et al. 2011

Commun. comput. phys.

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A moment method with closures based on Gaussian quadrature formulas is proposed to solve the Boltzmann kinetic equation with a hard-sphere collision kernel for mono-dispersed particles. Different orders of accuracy in terms of the moments of the velocity distribution function are considered, accounting for moments up to seventh order. Quadrature-based closures for four different models for inelastic collisionthe Bhatnagar-GrossKrook, ES-BGK, the Maxwell model for hard-sphere collisions, and the full Boltzmann hard-sphere collision integral-are derived and compared. The approach is validated studying a dilute non-isothermal granular flow of inelastic particles between two stationary Maxwellian walls. Results obtained from the kinetic models are compared with the predictions of molecular dynamics (MD) simulations of a nearly equivalent system with finite-size particles. The influence of the number of quadrature nodes used to approximate the velocity distribution function on the accuracy of the predictions is assessed. Results for constitutive quantities such as the stress tensor and the heat flux are provided, and show the capability of the quadrature-based approach to predict them in agreement with the MD simulations under dilute conditions. Abstract. A moment method with closures based on Gaussian quadrature formulas is proposed to solve the Boltzmann kinetic equation with a hard-sphere collision kernel for mono-dispersed particles. Different orders of accuracy in terms of the moments of the velocity distribution function are considered, accounting for moments up to seventh order. Quadrature-based closures for four different models for inelastic collisionthe Bhatnagar-Gross-Krook, ES-BGK, the Maxwell model for hard-sphere collisions, and the full Boltzmann hard-sphere collision integral-are derived and compared. The approach is validated studying a dilute non-isothermal granular flow of inelastic particles between two stationary Maxwellian walls. Results obtained from the kinetic models are compared with the predictions of molecular dynamics (MD) simulations of a nearly equivalent system with finite-size particles. The influence of the number of quadrature nodes used to approximate the velocity distribution function on the accuracy of the predictions is assessed. Results for constitutive quantities such as the stress tensor and the heat flux are provided, and show the capability of the quadrature-based approach to predict them in agreement with the MD simulations under dilute conditions. Disciplines Aerospace Engineering | Biological Engineering | Chemical Engineering | Mechanical Engineering Comments This article is from Communication in Computational Physics

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Section: Quadrature-based Moment Methodsmentioning

confidence: 99%

“…For the remaining moments, Eq. (3.3) provides an optimal approximation in the sense of Gaussian quadrature [27,33,37].…”

Section: Quadrature-based Moment Methodsmentioning

confidence: 99%

See 1 more Smart Citation

A Quadrature-Based Kinetic Model for Dilute Non-Isothermal Granular Flows

Passalacqua

Galvin²,

Vedula

et al. 2011

Commun. comput. phys.

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“…Specifically, we need the first 2m moments to construct recursively the orthogonal polynomials up to degree m. Neither Stiefel nor Rutishauser seem to ever mention this tool, despite the fact that, in its initial phase, the Rutishauser qd algorithm serves the same purpose, as we will see. The Chebyshev algorithm was later revived, analysed and modified by Sack & Donovan (1972), Wheeler (1974) and in a series of papers by Gautschi (the first of these being Gautschi, 1970), who also came up with the name modified Chebyshev algorithm for the more stable version using modified moments. Rutishauser (1954b) was aware of the work of Hadamard (1892), Aitken (1926Aitken ( , 1931 and Lanczos (1950) when he worked on Stiefel's problem.…”

Section: Finding the Poles Of F From The Moments: Hadamard And Aitkenmentioning

confidence: 99%

From qd to LR, or, how were the qd and LR algorithms discovered?

Gutknecht

Parlett

2010

IMA Journal of Numerical Analysis

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Perhaps, the most astonishing idea in eigenvalue computation is Rutishauser's idea of applying the LR transform to a matrix for generating a sequence of similar matrices that become more and more triangular. The same idea is the foundation of the ubiquitous QR algorithm. It is well known that this idea originated in Rutishauser's qd algorithm, which precedes the LR algorithm and can be understood as applying LR to a tridiagonal matrix. But how did Rutishauser discover qd and when did he find the qd-LR connection? We checked some of the early sources and have come up with an explanation.

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“…a very elegant algorithm due to Wheeler 22 can be used to evaluate the a's and b's in terms of the α's and β's and the modified moments µ i . Details of Wheeler's algorithm can be found in refs.…”

Section: Rys Quadraturementioning

confidence: 99%

Efficient electronic integrals and their generalized derivatives for object oriented implementations of electronic structure calculations

Flocke

Lotrich

2008

J Comput Chem

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Abstract:For the new parallel implementation of electronic structure methods in ACES III (Lotrich et al., in preparation) the present state-of-the-art algorithms for the evaluation of electronic integrals and their generalized derivatives were implemented in new object oriented codes with attention paid to efficient execution on modern processors with a deep hierarchy of data storage including multiple caches and memory banks. Particular attention has been paid to define proper integral blocks as basic building objects. These objects are stand-alone units and are no longer tied to any specific software. They can hence be used by any quantum chemistry code without modification. The integral blocks can be called at any time and in any sequence during the execution of an electronic structure program. Evaluation efficiency of these integral objects has been carefully tested and it compares well with other fast integral programs in the community. Correctness of the objects has been demonstrated by several application runs on real systems using the ACES III program.

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Modified moments and Gaussian quadratures

Cited by 182 publications

References 9 publications

A Quadrature-Based Kinetic Model for Dilute Non-Isothermal Granular Flows

A Quadrature-Based Kinetic Model for Dilute Non-Isothermal Granular Flows

From qd to LR, or, how were the qd and LR algorithms discovered?

Efficient electronic integrals and their generalized derivatives for object oriented implementations of electronic structure calculations

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