@article {5033,
title = {A spectral solution of the Sturm-Liouville equation: comparison of classical and nonclassical basis sets},
journal = {Journal of Computational and Applied Mathematics},
volume = {136},
number = {1-2},
year = {2001},
note = {ISI Document Delivery No.: 475METimes Cited: 16Cited Reference Count: 28},
month = {Nov},
pages = {17-35},
type = {Article},
abstract = {This paper considers a spectral method of solution of the Stunn-Liouville equation and the associated Schroedinger equation. The main objective is to develop a collocation method based on quadrature (collocation) points generated from nonclassical polynomials. The polynomials and associated quadrature points are calculated with Gautschi{\textquoteright}s Stieltjes procedure from some specified weight function. The particular spectral method used here with nonclassical basis sets is referred to as the Quadrature Discretization Method (QDM). The QDM and a related weighted QDM are applied to several Sturm-Liouville and Schroedinger equations and the results are compared with the traditional spectral methods based on Chebyshev and Legendre quadrature points. The results are also compared with the results of other workers wherever available. The QDM was found to give the most rapid convergence relative to other methods for the problems studied. (C) 2001 Elsevier Science B.V. All rights reserved.},
keywords = {LAGRANGE, MESHES, METHOD QDM, nonclassical polynomials, NUMERICAL-INTEGRATION, ORTHOGONAL POLYNOMIALS, QUADRATURE DISCRETIZATION METHOD, SCHRODINGER-EQUATION, Schroedinger equation, spectral methods, Sturm-Liouville equation},
isbn = {0377-0427},
url = {://000171167400002},
author = {Chen, H. and Shizgal, B. D.}
}
@article {4757,
title = {A direct spectral collocation Poisson solver in polar and cylindrical coordinates},
journal = {Journal of Computational Physics},
volume = {160},
number = {2},
year = {2000},
note = {ISI Document Delivery No.: 315BRTimes Cited: 22Cited Reference Count: 33},
month = {May},
pages = {453-469},
type = {Article},
abstract = {In this paper, we present a direct spectral collocation method for the solution of the Poisson equation in polar and cylindrical coordinates. The solver is applied to the Poisson equations for several different domains including a part of a disk, an annulus, a unit disk, and a cylinder. Unlike other Poisson solvers for geometries such as unit disks and cylinders, no pole condition is involved for the present solver. The method is easy to implement, fast, and gives spectral accuracy. We also use the weighted interpolation technique and nonclassical collocation points to improve the convergence. (C) 2000 Academic Press.},
keywords = {ARBITRARY ORDER ACCURACY, coordinates, cylindrical, EXPANSION, METHOD QDM, NONCLASSICAL BASIS FUNCTIONS, Poisson solver, polar coordinates, QUADRATURE DISCRETIZATION METHOD, SCHRODINGER-EQUATION, SINGULARITIES, spectral collocation, TSCHEBYSCHEFF POLYNOMIALS},
isbn = {0021-9991},
url = {://000087093200002},
author = {Chen, H. L. and Su, Y. H. and Shizgal, B. D.}
}
@article {4131,
title = {The quadrature discretization method in the solution of the Fokker-Planck equation with nonclassical basis functions},
journal = {Journal of Chemical Physics},
volume = {107},
number = {19},
year = {1997},
note = {ISI Document Delivery No.: YG655Times Cited: 12Cited Reference Count: 122},
month = {Nov},
pages = {8051-8063},
type = {Review},
abstract = {Fokker-Planck equations are used extensively to study a variety of problems in nonequilibrium statistical mechanics. A discretization method referred to as the quadrature discretization method (QDM) is introduced for the time-dependent solution of Fokker-Planck equations. The QDM is based on the discretization of the probability density function on a grid of points that coincide with the points of a quadrature. The quadrature is based on a set of nonclassical polynomials orthogonal with respect to some weight function. For the Fokker-Planck equation, the weight functions that have often provided rapid convergence of the eigenvalues of the Fokker-Planck operator are the steady distributions at infinite time. Calculations are carried out for several systems with bistable potentials that arise in the study of optical bistability. reactive systems and climate models. The rate of convergence of the eigenvalues and the eigenfunctions of the Fokker-Planck equation is very rapid with this approach. The time evolution is determined in terms of the expansion of the distribution function in the eigenfunctions. (C) 1997 American Institute of Physics.},
keywords = {CLIMATIC TRANSITIONS, COLORED NOISE, DISCRETE-ORDINATE METHOD, INITIAL-VALUE PROBLEM, ONE-DIMENSIONAL, OPTICAL BISTABILITY, ORTHOGONAL POLYNOMIALS, SCHRODINGER-EQUATION, STOCHASTIC RESONANCE, SYSTEMS, TIME-DEPENDENT NUCLEATION},
isbn = {0021-9606},
url = {://A1997YG65500053},
author = {Shizgal, B. D. and Chen, H. L.}
}
@article {4130,
title = {The quadrature discretization method (QDM) in the calculation of the rotational-vibrational transitions in rare gas dimers},
journal = {Theochem-Journal of Molecular Structure},
volume = {391},
number = {1-2},
year = {1997},
note = {ISI Document Delivery No.: WU757Times Cited: 3Cited Reference Count: 38},
month = {Feb},
pages = {131-139},
type = {Article},
abstract = {A discretization method referred to as the Quadrature Discretization Method (QDM) is employed for the determination of the rotational-vibrational states of the inert gas dimers. The QDM is based on the discretization of the wave function on a grid of points that coincide with the points of a quadrature. The quadrature is based on a set of non-classical polynomials orthogonal with respect to a weight function determined by the potential function in the Schroedinger equation. In this paper, the weight functions used in the solution of the Schroedinger equation are related to the ground state wavefunctions of the Morse potential that approximates the potential of interest. Calculations are carried out for vibrational energy levels for HF modelled with a Morse potential. Excellent agreement is obtained with the known spectrum of the Morse potential. The rate of convergence of the eigenvalues and the eigenfunctions of the Schroedinger equation is very rapid with this approach. The vibrational and rotational transitions for the rare gas dimers are calculated and compared with the results reported recently by Ogilvie and Wang (J. Mol. Struc., 291 (1993) 313). (C) 1997 Elsevier Science B.V.},
keywords = {AR-XE, COEFFICIENTS, DISCRETE-VARIABLE REPRESENTATION, NE-KR, ORDINATE METHOD, POTENTIALS, quadrature discretion method, QUANTUM-MECHANICS, rare gas dimer, rotation-vibration, SCHRODINGER-EQUATION, Schroedinger equation, SPECTRA, STATES, TRANSITION},
isbn = {0166-1280},
url = {://A1997WU75700014},
author = {Shizgal, B. D.}
}