Finite difference calculation method

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. 

At this point, specification of the finite-difference solution method is complete in that we have chosen to utilize the finite-difference scheme and have specified the mesh properties and the sampling of points required to provide the desired approximation to the derivatives of U. Such systems can be solved efficiently even for N and M large (>1000), although time scales of typical calculations range from several minutes to hours, depending on the type of computers used, eliminating some from consideration in those time-critical situations. We defer additional discussion of achieving solutions until the next step. 

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. 

Potential energy descriptors proposed as an indicator of hydrophobicity [Oprea and Waller, 1997]. Originally, they were calculated using the finite difference approximation method the linearized Poisson-Boltzmann equation was solved numerically to compute the electrostatic contribution to solvation at each grid point. Desolvation energy field values were calculated as the difference between solvated (grid dielectric = 80) and in vacuo (grid dielectric = 1). 

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 ). 

In order to obtain some indication of the reliability of these variational results, calculations were made with 16 central functions at those energies where the permeabilities by the two methods were in greatest disagreement. The permeability obtained for such a calculation for reaction (1) at 12.5 kcal/ mole is 0.490, which is 0.005 larger than that obtained from the 25 central function calculation. This result is consistent with the permeabilities found in Table II for this reaction. For reaction (2) the permeability calculated using 16 central functions at 12.0 kcal/mole is 0.666, which is 0.013 larger than that listed in Table II. Even with a possible error of 0.01-0.02 in the variational calculations, there is still a small discrepancy between the two methods of calculation, which probably indicates that a somewhat similar error exists in the finite difference calculations for reaction (2). 

Finite difference calculation Parallel beam approximation Static response method.s... 

The advantage of the finite difference method is the simple computer implementation of the procedures and thus, it is easy to write ovra codes and to implement or consider new features. The drawback can be the consideration of the boundary conditions for complex shaped geometries and the consideration of the symmetry of the stiffiiess matrix might be difficult. Thus, many applications of the finite difference method are restricted to simple geometries. To overcome these problems, the so called finite difference energy method was developed (Bushnell et al. 1971) where the displacement derivatives in the total potential energy of a system are approximated by finite differences and the minimum condition of the potential energy is used to calculate the unknown displacements. 

Use METHOD = -1 for backward finite difference Use METHOD = 0 for central finite difference Use METHOD = 1 for forward finite difference ERR is the order of error of calculation. ERR may be 1 or 2 for backward and forward finite difference and 2 or 4 for central finite difference. 

The core neutronic calculation code used here is the COREBN code in SRAC2002. COREBN is a multidimensional core bumup calculation code based on macrocross section interpolations by bumups and the finite difference diffusion method. The macro-cross section sets required by the core bumup calculations can be prepared by numerous cell bumup calculations and assembly bumup calculations. 

As is briefly mentioned above, COREBN is based on the macro-cross section interpolations by bumups and the finite difference diffusion method for the neutron flux calculations. The macro-cross section sets for each fuel assembly type are prepared by ASMBURN as described above. COREBN linearly interpolates the macro-cross section sets tabulated for the three parameters, namely, the bumup, fuel temperature, and the moderator temperature. The bumup process of COREBN is similar to that of ASMBURN. Since COREBN is not equipped with a coupling function to the thermal-hydraulic calculations, the user has to give the input data of fuel temperature and moderator temperature for the calculations. 

The gradient of the PES (force) can in principle be calculated by finite difference methods. This is, however, extremely inefficient, requiring many evaluations of the wave function. Gradient methods in quantum chemistiy are fortunately now very advanced, and analytic gradients are available for a wide variety of ab initio methods [123-127]. Note that if the wave function depends on a set of parameters X], for example, the expansion coefficients of the basis functions used to build the orbitals in molecular orbital (MO) theory. 

Because of the use of the focusing method [18], more than four calculations are actually carried out for each group. However, the focusing method saves computer time by permitting the use of less extensive finite-difference grids. 

Use of the finite-difference PB (FDPB) method to calculate the self- and interaction-energies of the ionizable groups in the protein and solvent. 

There are many algorithms for integrating the equations of motion using finite difference methods, several of which are commonly used in molecular dynamics calculations. All algorithms assume that the positions and dynamic properties (velocities, accelerations, etc.) can be approximated as Taylor series expansions ... 

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