Finite difference calculation

One of the simplest ways to model polymers is as a continuum with various properties. These types of calculations are usually done by engineers for determining the stress and strain on an object made of that material. This is usually a numerical finite element or finite difference calculation, a subject that will not be discussed further in this book. 

Holen, J., M. Brostrom, and B. F. Magnussen, 1990, Finite Difference Calculation of Pool Fires, Pmc of 23rd Int. Symp, Combustion, pp. 1677-1683. 

Thermochemistry. Chen et al.168 combined the Kohn-Sham formalism with finite difference calculations of the reaction field potential. The effect of mobile ions into on the reaction field potential Poisson-Boltzman equation. The authors used the DFT(B88/P86)/SCRF method to study solvation energies, dipole moments of solvated molecules, and absolute pKa values for a variety of small organic molecules. The list of molecules studied with this approach was subsequently extended182. A simplified version, where the reaction field was calculated only at the end of the SCF cycle, was applied to study redox potentials of several iron-sulphur clusters181. 

Although convection, axial diffusion, and radial diffusion actually occur simultaneously, a multistep procedure was adopted in the finite-difference calculation. For each 5-cm increment in tidal volume and for each time increment At, the differential mass-balance equations were solved for convection, axial difihision, and radial diffusion in that order. This method may slightly underestimate the dosage for weakly soluble gases, because the concentration gradient in the airway may be decreased. 

The example of a CO molecule has shown that the displacements used in finite-difference calculations for vibrational frequencies should be not too small and not too large. In practice, it is a good idea to choose displacements that result in energy differences on the order of 0.01-0.10 eV since these energy differences can be calculated accurately without requiring extraordinary... 

Section 6.2 Frequency calculations were performed on the Ag adatom on the Cu(100) hollow and bridge sites using 4x4x1 k points. The frequencies were determined via finite-difference calculations using a displacement of 0.04 A. 

Pleshanov (P4) extends the integral heat balance method to bodies symmetric in one, two, or three dimensions, using a quadratic polynomial for the approximate temperature function. Solutions are obtained in terms of modified Bessel functions which agree well with numerical finite-difference calculations. 

Holen, J., Brostrom, M., and Magnussen, B.F. Finite difference calculation of pool fires. In Proceedings of the 23rd Symposium (International) on Combustion. Pittsburgh, PA Combustion Institute, 1990, pp. 1677-1683. 

Hunter I.C., Jones I.P., Numerical experiments on the effects of strong grid stretching in finite difference calculations, Tech. Rep. AERE R-10301, United Kingdom Atomic Energy Authority, Harwell (1981)... 

Table 3-2 Summary of Nodal Formulas for Finite-Difference Calculations (Dashed Lines Indicate Element Volume.)...

Maximum temperatures measured in the adsorbent during the adsorption of nitrogen on 4A and propane on 5A zeolite, both at —78°C, were 15° and 50°C above the bath temperature. Finite difference calculations, taking into account the generation and loss of heat and changes in diffusivity and equilibrium adsorption with temperature, reproduced the pertinent features of the rate and temperature data. When the temperature maximum occurs late in the adsorption process, the rate curve is drastically different from that expected for isothermal adsorption. 

Analytical gradient calculations are quite effective when compared to the finite difference calculations. A factor of 20 applies in the present PCM formulation (Cossi et al., 1995). Although effective, this increment of efficiency is smaller, by a factor 10 or more, than the analogous speed up found for the calculations of gradients in vacuo. 

To be acceptable, a calculation procedure must be stable, and the amount of truncation error introduced must be the one required. In this subsection, we evaluate the stability condition of each of the finite difference calculation procedures described above, and we determine the amount of truncation error done with each of these schemes. However, because of the great difficulty of these theoretical studies when handling the nonlinear case, we limit this discussion of the stability condition and the truncation error made with the three schemes we have described to the case of a linear isotherm. 

Equilibrium-stage processes are discrete steps. One approach to the analysis is an evaluation as a finite difference calculation where each stage is an equal and discrete interval in the process train. Obviously, a process simulator can be used. Finite difference approaches, including ones shown here, can be implemented on spreadsheets for rapid estimates. 

Table 4.1 Exchange-only ground-state energies from ROPM and RHF calculations for noble gas atoms Coulomb (C) and Coulomb-Breit (C + B) limit in comparison with complete transverse exchange (C + T) (Engel et al. 1998a). For the RHF approximation the energy difference with respect to the ROPM is given, AE = tot(RHF) — tot(ROPM), providing results from (a) finite-differences calculations (Dyall et al. 1989) and (b) a basis-set expansion (Ishikawa and Koc 1994). All energies in mHartree. uext and c as in Ishikawa and Koc (1994).

So we can start with the known initial mass of water, then using the Steam Tables and the data at every 5°C do a finite difference calculation to obtain the results below. 

These results are similar to that from the energy balance. The differences are the result of round off errors in the simple finite difference calculation scheme used here (i.e., more complicated predictor-corrector methods would yield more accurate results ). 

Finite-Difference Calculations for Atoms and Diatomic Molecules in Strong Magnetic and Static Electric Fields... 

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