Numerical grid

C. Cerjan, ed., Numerical Grid Methods and their Application to Schrodinger s Equation, Kluwer Academic Publishers, Dordrecht, 1993,... 

J.F. Thompson, Z.U.A. Warsi, and C.W. Mastin, Numerical Grid Generation, North-Holland, Amsterdam, 1985. 

Molecules and Clusters. The local nature of the effective Hamiltonian in the LDF equations makes it possible to solve the LDF equations for molecular systems by a numerical LCAO approach (16,17). In this approach (17), the atomic basis functions are constructed numerically for free atoms and ions and tabulated on a numerical grid. By construction, the molecular basis becomes exact as the system dissociates into its atoms. The effective potential is given on the same numerical grid as the basis functions. The matrix elements of the effective LDF Hamiltonian in the atomic basis are given by... 

There are several improvements that can be made to the present calibration procedure. First of all the numerical grid in the DST model can be adjusted to yield more accurate results for short time step data. Secondly the accuracy of the calibration can be increased. At present error for any time step of the calibrated temperatures is A0.2-0.4 K, this should probably be decreased to about 0.1-0.2 K. 

We illustrated how these equations are discretized over an appropriate numerical grid and also showed some sample results. One can readily appreciate that one must choose the grid sizes in the numerical solution of the TFM equations to be smaller than the shortest length scale at which the TFM equations afford inhomogeneities. This requirement leads to a practical difficulty when one tries to solve these microscopic TFM equations for gas-particle flows, as discussed below. 

G. C. Corey, J. W. Tromp, and D. Lemoine, in Numerical Grid Methods and Their... 

G. G. Balint-Kurti and A. Vibok, Complex absorbing potentials in time dependent quantum dynamics, in Numerical Grid Methods and Their Application to Schrdinger s Equation, C. Ceijan, ed., NATO ASI series, Series C Mathematical and Physical Sciences, Vol. 412 (Kluwer Academic Publishers, 1993), p. 195. 

T.N. Rescigno, C.W. McCurdy, Numerical grid methods for quantum-mechanical scattering problems, Phys. Rev. A 62 (3) (2000) 032706. 

To avoid dewatering the aquifer, constraints can be imposed on either the upper bounds of extraction rates (e.g., Rastogi 1989 Marryott et al. 1993) or the minimum saturated thickness at numerical grid cells (e.g., Ahlfeld et al. 1998). The later approach pushes the physical limits of the system by implicitly allowing the system behavior to define the upper pumping bounds. In this later approach, however, care must be taken to ensure that the optimization search procedure remains feasible with respect to the saturated thickness constraints. 

A common approach in three-dimensional groundwater simulation (e g., McDonald and Harbaugh 1984) is to represent the vertical dimension with several numerical grid layers and to determine the head within each layer using equation (1). Flow between the layers is accommodated with a vertical conductance term. 

During simulation of unconfined aquifers, care must be exercised to avoid dewatering numerical grid cells because of excessive pumping. If extraction well locations are assumed to be most prone to dewatering then either upper bounds on extraction rates or lower bounds on head at the well cell can be imposed on the pumping solution. Imposing explicit upper bounds on extraction may artificially eliminate the best solutions from consideration. With this in mind, the approach taken here is to impose lower... 

So far we have only discussed harmonic frequencies. The effect of anhar-monicity can be treated using either a Taylor expansion of the PES in terms of normal mode coordinates or by explicitly spanning the PES on a numerical grid. The discussion of anharmonic force constants is postponed to the following section. Here, we will focus on an explicit PES generated by means of the following correlation expansion, here written up to three-mode correlations, [53]... 

D.C. Ives. Conformal grid generation. In Numerical Grid Generation, page 107, New York, 1982. Elsevier. 

Our AIMD simulations are all-electron and self-consistent at each 0.4 femtoseconds (fs) time step. Variational fitting ensures accurate forces for any finite orbital or fitting basis sets and any finite numerical grid. These forces are used to propagate the nuclear motion according to the velocity Verlet algorithm [22]. The accuracy of these methods is indicated by the fact that during the 500 time-step simulations of methyl iodide dissociation described below, the center of mass moved by less than 10-6 A. 

Matrix Assembler (ADMA) method [18-21] generates a macromolecular density matrix P((p(K)) that can be used for the computation of a variety of molecular properties besides ab initio quality macromolecular electron densities. In electron density computations the accuracy of the ADMA macromolecular density matrix P(cp(K)) corresponds to that of a MEDLA result of an infinite resolution numerical grid. 

The procedure chosen to calculate E nt must ensure that electronic energies of the dimer and monomers are evaluated in a consistent manner [19,21-24], It should be stressed that this requirement is absolutely crucial, as no method at present can in practice yield EAg, Ea and Eg energies with an absolute error smaller than E nt. Therefore, Eq. (2), which defines the interaction energy does not offer so simple a computational approach as might be expected at first glance. Two notorious inconsistencies to be alleviated in practice are basis set inconsistency (same basis set expansion or numerical grid for A, B, and AB must be used, otherwise the basis set superposition error (BSSE) arises [21,23,24]) and the size inconsistency (a theory to describe AB must guarantee a correct dissociation into A and B, at the same level of theory [25]). 

The M D Group also develops the CERA-2 data model for the World Climate Data Centre, proposing a description of geo-referenced climate data (model output) and containing information for the detection, browse and use of data. An important collaboration is going on with the PAE Metadata and other international initiatives for the development and implementation of metadata standards for the description of model configuration and numerical grids. 

Numerical grid metadata, developed by Balaji at GFDL, USA, for numerical grid description (http //www.gfdl.noaa.gov/ vb/gridstd/gridstd.html). 

Thompson, J. E, Warsi, Z. U. A., and Mastin, C. W., Numerical Grid Generation, Foundations and Applications. North-Holland, New York, 1985. 

R. Kosloff, The fourier method. In C. Ceijan, editor. Numerical Grid Methods and Their Application to Schr6dinger s Equation, volume NATO ASI Ser. C 412, 175-194, Kluwer Academic Publishers, The Netherlands, (1993). 

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