A discrete ordinate method of solution of linear boundary value and eigenvalue problems

Shizgal, Bernie D.; Blackmore, R.

doi:10.1016/0021-9991(84)90009-3

Cited by 96 publications

(38 citation statements)

References 10 publications

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“…These speed polynomials are eigenfunctions for a Fokker-Planck operator (Shizgal and Blackmore, 1984). They form a complete basis of orthonormal functions: Z 1 0 y 2 expðÀy 2 ÞS s ðyÞS r ðyÞdy ¼ d sr , (C.2)…”

Section: Appendix C Polynomial Expansion To Solve the Fokker-planck mentioning

confidence: 99%

Kinetic modeling of the polar wind

Tam

Chang

Pierrard

2007

Journal of Atmospheric and Solar-Terrestrial Physics

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Section: Appendix C Polynomial Expansion To Solve the Fokker-planck mentioning

confidence: 99%

Kinetic modeling of the polar wind

Tam

Chang

Pierrard

2007

Journal of Atmospheric and Solar-Terrestrial Physics

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“…As shown by Dickinson and Certain 28, the eigenvalues of the coordinate operator, x n. are the N Gaussian quadrature points for the Nth order quadrature appropriate to the functions , and the transformation matrices have elements (5) For real basis functions, T is an orthogonal matrix, and thus we have the "di scre te orthonormality· relations among the DVR functions on the quadrature : (6) This also leads to the "Kronecker dett a" property of the DVR basis :…”

Section: Gaussian Quaorature Ovr'smentioning

confidence: 99%

Discrete Variable Representations in Quantum Dynamics

Light

1992

Nato ASI Series

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A derivation of discrete variable representations (DVR's) is given, with emphasis on formal properties and relations to the classical orthogonal polynomials. These "Gaussian quadrature" DVR's are compared with several different DVR's proposed recently. Finally, some cautions and brief comparisons of accuracy are noted for quantum solutions for large amplitude motions of triatomic molecules (H 20 in this case).

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“…The numerical methods used previously (15,26) where N is the dimension of the matrix and the points, xi, and weights, wi, are associated with the quadraturediscr^etization method (QDM) (15,16) (the relationship between +(x) and +(x) is discussed in these previous papers). As a result (33) the initial condition is fitted exactly, that is, By contrast, the eigenfunctions determined with the WKB and (or) SWKB approaches are not required to satisfy a similar completeness relationship (nor are they necessarily orthogonal) so that the initial condition is not fitted exactlv.…”

Section: =0mentioning

confidence: 99%

“…11-14). A useful numerical procedure for the solution of such Fokker-Planck equations was introduced by Blackmore and Shizgal (15,16) and has been applied extensively. There is an important relationship between the Fokker-Planck eigenvalue problem and a corresponding Schrodinger equation with a potential function that belongs to the class of potentials that arise in supersymmetric quantum mechanics, as discussed elsewhere (14).…”

Section: Introductionmentioning

confidence: 99%

Comparison of WKB (Wentzel–Kramers–Brillouin) and SWKB solutions of Fokker–Planck equations with exact results; application to electron thermalization

Shizgal¹,

Demeio²

1991

Can. J. Phys.

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A comparison of WKB (Wentzel-Kramers-Brillouin) and SWKB eigenfunctions of the Schrodinger equation for potentials in the class encountered in supersymmetric quantum mechanics is presented. The potentials that are studied are those that result from the transformation of a Fokker-Planck eigenvalue problem to a Schrodinger equation. Linear Fokker-Planck equations of the type considered in this paper give the probability distribution function for a large number of physical situations. The time-dependent solutions can be expressed as a sum of exponential terms with each term characterized by an eigenvalue of the Fokker-Planck operator. The specific Fokker-Planck operator considered is the one that describes the thermalization of electrons in the inert gases. The WKB and SWKB semiclassical approximations are compared with exact numerical results. Although the eigenvalues can be very close to the exact values, we find significant departures for the eigenfunctions.On presente une comparaison entre les fonctions propres WKB (Wentzel-Kramers-Brillouin) et SWKB de I'Cquation de Schrodinger pour des potentiels de la classe rencontrke en mkcanique quantique supersymitrique. Les potentiels qui sont CtudiCs sont ceux qui rksultent de la transformation d'un problhe Fokker-Plank de valeurs propres en une equation de Schrodinger. Les equations Fokker-Plank IinCaires du type considere dans cet article donne la fonction de distribution de probabilite pour un grand nombre de situations physiques. Les solutions dipendantes du temps peuvent Ctre exprimkes comme une somme de termes exponentiels dont chacun est caractCrist par une valeur propre de I'opCrateur Fokker-Plank. L'opCrateur Fokker-Plank spkcifique considCrC est celui qui dCcrit la thermalisation d'electrons dans les gaz inertes. Les approximations semiclassiques WKB et SWKB son! comparCes avec les rtsultats nurntriques exacts. Bien que les valeurs propres approximative~ se rapprochent beaucoup des valeurs exactes, nous trouvons des Ccarts significatifs.[Traduit par la redaction]Can. J . Phys. 69. 712 (1991)

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A discrete ordinate method of solution of linear boundary value and eigenvalue problems

Cited by 96 publications

References 10 publications

Kinetic modeling of the polar wind

Kinetic modeling of the polar wind

Discrete Variable Representations in Quantum Dynamics

Comparison of WKB (Wentzel–Kramers–Brillouin) and SWKB solutions of Fokker–Planck equations with exact results; application to electron thermalization

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