Use Of Difference Approximations Research Articles

A Nodal-iterative Technique for Criticality Calculations in Multigroup Neutron Diffusion Models

In this work, a nodal and iterative technique to evaluate the effective multiplication factor as well as the neutron flux, in multigroup diffusion problems, is presented. An iterative scheme, similar to the source iteration method, is implemented to decouple the system of differential equations which is the fundamental mathematical model. Then, analytical solutions are derived for the one-dimensional transverse integrated equations, of each energy group, resulting from a nodal approach. Constant approximations are assumed for the unknown transverse leakage terms in the contours of the nodes. In addition, constant and linear representations are investigated to express the fluxes in the source term to be updated in the iterative process. Numerical results for the effective multiplication factor were obtained for a series of two-dimensional multigroup problems with up-scattering and down-scattering. The procedure is simple, fast, the analysis of the results indicated a satisfactory agreement with results available in the literature and the use of different approximations to the source term seems to be a good alternative, instead of using higher-order approximations on the contour of the nodes, to improve accuracy.

Estimation of H2O Absorption Line Contributions to Atmospheric Transmission in the Ultraviolet Spectral Region

The POKAZATEL line list is currently the most complete theoretical list of H2O monomer absorption lines. It contains H2O lines up to the dissociation energy ∼40 000 cm−1 (0.25 μm). The atmospheric transmission of ultraviolet (UV) radiation is simulated with the use of H2O lines from POKAZATEL. The broadening parameters of absorption lines in the UV region are determined more accurately with the use of different approximations. It is shown that the contribution of H2O absorption lines to the atmospheric transmission can attain 0.03 near 25 000 cm−1 at a spectral resolution of 0.01 cm−1.

On hybrid convolution quadrature approaches for modeling time‐domain wave problems with broadband frequency content

AbstractWe propose two hybrid convolution quadrature based discretizations of the wave equation on interior domains with broadband Neumann boundary data or source terms. The convolution quadrature method transforms the time‐domain wave problem into a series of Helmholtz problems with complex‐valued wavenumbers, in which the boundary data and solutions are connected to those of the original problem through the ‐transform. The hybrid method terminology refers specifically to the use of different approximations of these Helmholtz problems, depending on the frequency. For lower frequencies, we employ the boundary element method, while for more oscillatory problems, we develop two alternative high frequency approximations based on plane wave decompositions of the acoustic field on the boundary. In the first approach, we apply dynamical energy analysis to numerically approximate the plane wave amplitudes. The phases will then be reconstructed using a novel approach based on matching the boundary element solution to the plane wave ansatz in the frequency region where we switch between the low and high frequency methods. The second high frequency method is based on applying the Neumann‐to‐Dirichlet map for plane waves to the given boundary data. Finally, we investigate the effectiveness of both hybrid approaches across a range of numerical experiments.

Impact of nonconvergence and various approximations of the partition function on the molecular column densities in the interstellar medium

We emphasize that the completeness of the partition function, that is, the use of a converged partition function at the typical temperature range of the survey, is very important to decrease the uncertainty on this quantity and thus to derive reliable interstellar molecular densities. In that context, we show how the use of different approximations for the rovibrational partition function together with some interpolation and/or extrapolation procedures may affect the estimate of the interstellar molecular column density. For that purpose, we apply the partition function calculations to astronomical observations performed with the IRAM-30 m telescope towards the NGC 7538–IRS1 source of two N-bearing molecules: isocyanic acid (HNCO, a quasilinear molecule) and methyl cyanide (CH3CN, a symmetric top molecule). The case of methyl formate (HCOOCH3), which is an asymmetric top O-bearing molecule containing an internal rotor is also discussed. Our analysis shows that the use of different partition function approximations leads to relative differences in the resulting column densities in the range 9–43%. Thus, we expect this work to be relevant for surveys of sources with temperatures higher than 300 K and to observations in the infrared.

Space–time proper generalized decompositions for the resolution of transient elastodynamic models

In this paper, we investigate ability of proper generalized decomposition (PGD) to solve transient elastodynamic models in space–time domain. Classical methods use time integration schemes and an incremental resolution process. We propose here to use standard time integration methods in a non-incremental strategy. As a result, PGD gives a separated representation of the space–time solution as a sum of tensorial products of space and time vectors, that we interpret as space–time modes. Recent time integration schemes are based on multi-field formulations. In this case, separated representation can be constructed using state vectors in space and same vectors in time. However, we have experienced bad convergence order using this decomposition. Furthermore, temporal approximation must be the same for all fields. Thus, we propose an extension of classical separated representation for multi-field problems. This multi-field PGD (MF-PGD) uses space and time vectors that are different for each field. Calculation of decomposition is done using a monolithic approach in space and time, potentially allowing the use of different approximations in space and time. Finally, several simulations are performed with the transient elastodynamic problem with one dimension in space. Different approximations in time are investigated: Newmark scheme, single field time discontinuous Galerkin method and two fields time continuous and discontinuous Galerkin methods.

On the Approximation Bias to Benefit Measures in Discrete Choice Models

Bias estimation associated with use of different approximations for transportation project benefit calculation is presented by the authors. A simulated databank was generated by the authors for real revealed preference sample behavior reproduction on work trip mode choice. There was exact measure computation for three transportation policies differing in mode choice impact degree. Simpler but less precise measure application results were then compared against these values, showing significant bias existence with classical explanatory model consideration. Correct methodology application for marginal changes in the presence of non-marginal policies also detected significant biases. Proportional error, in the first case, does not depend on policy size, but, in the second case, grows with their impact. There was another databank that the authors also simulated with non-linear income utilities and some welfare measures were again calculated. That no significant biases exist if there is no consideration of income effects was found by the authors, as was that there is no systematic relation between policy size (in terms of change) and these biases.

Theoretical Study on Non-local Optical Potential Applied to XPD.

In accurate analyses of XPD spectra, the atomic potential at each site plays an important role. Hermitian potential like Hartree-Fock potential cannot describe the damping of the photoelectron wave, so that it is crucial to go beyond one-electron theory for the XPD analyses. Our optical potential is energy dependent, non-local and self-consistent based on Hedin's GW approximation. Here we apply this optical potential to the N 1s XPD spectra of N2 adsorbed on metals, and compare the results by use of different approximations.

Electron-phonon interaction and low-field drift mobility in a polar semiconductor quantum well

We have calculated relaxation times by means of a general iterative procedure when the electron-phonon interaction is included via three scattering mechanisms: the acoustic deformation potential, the piezoelectric of the acoustic branch, and longitudinal optical mode coupling (LO). Confinement of LO vibrations is taken into account by means of an appropriate Hamiltonian, and this leads to some selection rules for this interaction, which are carefully discussed. Screening of the electron-phonon interaction is considered. Mobility calculations are carried out in a size-quantum-limit (SQL) approximation. We have compared the effects on the screening contributions of the different scattering mechanisms and the confinement effects, produced by the use of different approximations for the electronic part, ranging from a self-consistent solution to an infinite barrier model.

Thermodynamic properties and crystallization kinetics of glass-forming undercooled liquid Au-Pb-Sb alloys

The heat capacities of liquid and crystalline Au-Pb-Sb alloys in the glass-forming composition range were measured with droplet emulsion and bulk samples. Based on the measured Cp data, the entropy, enthalpy, and Gibbs free-energy differences between the eutectic solid mixture and undercooled liquid were determined as a function of temperature over ∼60% of the undercooling range below the liquidus temperature and compared with theoretical predictions. The results indicate an isentropic temperature at 313 (±5) K, which agrees well with experimental data for the glass transition. The thermodynamic evaluation was applied further to develop a kinetics analysis of the nucleation undercooling response during cooling. Use of different approximations for the Gibbs free energy leads to a variation of the prefactor terms of six orders of magnitude for classical nucleation theory and, consequently, large variation in calculated transformation diagrams which is more pronounced with increasing undercooling. Extrapolations into the glass-forming temperature range and the effects of viscosity, transient nucleation, and estimated Kauzmann temperatures on the crystallization kinetics at high undercooling have been evaluated. This analysis reveals the importance of using measured values of thermophysical properties, even if they represent a limited temperature range at modest undercooling, rather than model approximations in order to obtain reliable evaluations of crystallization kinetics at high undercooling in the glass-forming temperature range.

The representation of time variant free surface seepage using finite difference methods: a review

Abstract Several numerical techniques are available for the representation of time variant free surface seepage, notably finite elements, boundary elements and finite difference methods. This work concentrates on the latter. Over the last twenty years several papers have been published on the representation of free surface seepage using finite differences but unfortunately each paper advocates the use of different approximations and stability criteria. The choice of which variation to use is therefore a problem. In this paper the various methods are reviewed and tested against a variety of solutions and recommendations are made as to which method is the most stable and accurate.

On the construction of K-consistent difference schemes of gas dynamics

Previously it was found by the present authors that the use of non-classical equations of state in gas dynamic computations by Eulerian difference schemes may give rise to spurious oscillations of the numerical solution in the vicinity of contact discontinuities. Two accurate methods for elimination of these oscillations are proposed below. One of these methods is based on a new definition of the K-consistency of difference schemes. Another method makes use of difference approximations of an unsteady equation for pressure which is a well-known consequence of the Euler equations. The effectiveness of the methods proposed is illustrated by computational results.

A Numerical Model Study of Gravitational Effects and Production Rate on Solution Gas Drive Performance of Oil Reservoirs

Ultimate oil recovery by solution gas drive can be greatly affected by variations in production rates --- in general the higher the rate the higher the recovery. However, the quantitative relationship for a given field will be influenced by factors other than rate and vertical permeability: reservoir thickness, gas solubility, shrinkage factor, oil permeability: reservoir thickness, gas solubility, shrinkage factor, oil viscosity, relative permeability, and capillary pressure. Introduction The relationship between ultimate oil recovery and rate of oil production has been the subject of many investigations since 1924 when Cutler published a study of field data. Cutler concluded that higher oil production rates yielded higher ultimate oil production rates yielded higher ultimate oil recoveries. Subsequent studies of performance from a variety of fields showed that factors other than producing rate caused so much variation in total oil producing rate caused so much variation in total oil recovered that no conclusion could be reached concerning the effects attributable to this single factor. Because of the impossibility of imposing different production histories on a given reservoir to observe production histories on a given reservoir to observe the difference in ultimate oil produced, confirmation of any such relationship had to depend on analytical studies and laboratory model tests. Analytical approaches using simplified models, in which effects of capillary pressure and gravity forces were ignored, led to the conclusion that any effect of production rate upon ultimate recovery was negligible. They did show gross differences in horizontal saturation distributions imposed by production rate variations. In 1953 and 1954 the foundation was laid for solution of the unsteady-state fluid flow equations by use of difference approximations and high-speed computers. The development of these techniques led to numerous mathematical model investigations of fluid displacement in one and two dimensions. These works treated a range of fluid systems in one and two dimensions; generally they were confined to incompressible fluids, and included gravity and capillary, effects. When simulating compressible fluid systems, gravity and capillary forces were ignored. Levine et al. and Heuer et al. studied the effects of production rate on ultimate oil recovery by solution gas drive, using these more sophisticated mathematical techniques. Both efforts neglected the effect of gravity, but Levine included capillary forces. Both concluded that producing rate had a negligible effect on ultimate recovery. Ridings et al. showed that performance of long horizontal laboratory systems performance of long horizontal laboratory systems coincided closely with performance calculated by numerical modeling techniques, with one exception. Noticeably increasing recovery was observed with increasing production rate in the laboratory system, whereas the numerical model showed no rate effect. Gravity was not a factor in either system. Since the production rate sensitivity was less on very long production rate sensitivity was less on very long laboratory systems than for the shorter ones, it was concluded that the rate effect observed on the short laboratory system was, for reasons not specified, typical only of short systems. It bas been recognized for some time that, under certain conditions, gravitational forces can be very important in the displacement of oil from natural reservoirs. Muskat treats exhaustively the performance of wells and reservoirs in which gravity is performance of wells and reservoirs in which gravity is the sole driving mechanism. JPT P. 625

Solution of a non-linear discrete boundary value problem by a continuous analog of Newton's method

IN this paper we consider the non-linear discrete boundary value problem obtained from a finite difference approximation of the boundary value problem for the ordinary differential equation ϑ = y″ + ƒ(x, y) = 0 (0.1) y(0) = y(1) = 0 (0.2) In [1] a continuous analog of Newton's method was used to find an approximate solution of problem (0.1)–(0.2). The existence of these solutions is assumed. The proposed computational scheme is a realization of Euler's method of solving a Cauchy problem for a certain differential equation in B-space. It reduces to the solution for a fixed value of the parameter t of a boundary value problem for a linear ordinary differential equation, and to a subsequent advancement along the parameter by a simple difference relation. The solution of the boundary value problem for fixed t is assumed to be found numerically, for example, by the use of difference approximations. It is easy to see that this computational scheme is a numerical realization of the continuous analog of Newton's method of solving the difference operator equation approximating to the original boundary value problem (0.1)–(0.2). In the present paper a justification of this method is given: a proof is given of the existence of a solution of the discrete boundary value problem which is the difference analog of problem (0.1)–(0.2), and the convergence of the approximate solution to the solution of the original problem is proved. The result obtained in [2] is used in proving the existence of the solution of the discrete boundary value problem. Similar problems were considered, for example, in [3], pp. 347–388, [4], pp. 59–67. Thus, in [4] a proof is given of the existence and method of finding an approximate solution of problem (0.1)–(0.2) by Newton's method. However, by connecting the questions of the existence and uniqueness of the solution of the original problem with the method of the approximate solution, the author obtains extremely restricted conditions for the applicability of the method.