Scalar approximation

To approximate scalar grid cell variables at the staggered velocity grid cell surface points, arithmetic interpolation is used ... 

To approximate scalar grid cell variables at the staggered w-velocity grid cell center node point, arithmetic interpolation is frequently used. The radial velocity component is discretized in the staggered t -grid cell volume and need to be interpolated to the w-grid cell center node point. The derivatives of the w-velocity component is approximated by a central difference scheme. When needed, arithmetic interpolation is used for the velocity components as well. 

The different approximations for all-electron relativistic calculations using one-component methods have recently been compared with each other and with relativistic ECP calculations of TM carbonyls by several workers (47,55). Table 6 shows the calculated bond lengths and FBDEs for the group 6 hexacarbonyls predicted when different relativistic methods are used. The results, which were obtained at the nonrelativistic DFT level, show the increase in the relativistic effects from 3d to 4d and 5d elements. It becomes obvious that the all-electron DFT calculations using the different relativistic approximations—scalar-relativistic (SR) zero-order regular approximation (ZORA), quasi-relativistic (QR) Pauli... 

Expression (21) is again a convenient starting point for further approximations. Scalar and spin-orbit contributions can be separated and by omitting the spin-orbit contributions, one-component quasi-relativistic models are obtained. 

We may be able to make some progress in solving the ABC system by appealing to the Newton polynomial expansion, which is a useful method for approximating scalar functions. A special case is the Chebyshev polynomial expansion, which is widely used in commercial algorithms for evaluating special functions [37, 38]. Based on the Chebyshev expansion for u x) = for x real, Tal-Ezer and Kosloff [5]... 

The approximation of Fresnel is scalar approximation. Let u(, r],0-0) be the scalar wave function of the laser beam falling onto the optical element, and u( X,y,Cl) will the be scalar wave function in the plane Z = Cl. Then [3,4]... 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

The errors in the present stochastic path formalism reflect short time information rather than long time information. Short time data are easier to extract from atomically detailed simulations. We set the second moment of the errors in the trajectory - [Pg.274]

Approximating the nonlinear force f(y) over a time step by a suitable constant vector leads to a scheme whose origins for scalar equations can be traced back to [10] ... 

This method, because it involves minimizing the sum of squares of the deviations xi — p, is called the method of least squares. We have encountered the principle before in our discussion of the most probable velocity of an individual particle (atom or molecule), given a Gaussian distr ibution of particle velocities. It is ver y powerful, and we shall use it in a number of different settings to obtain the best approximation to a data set of scalars (arithmetic mean), the best approximation to a straight line, and the best approximation to parabolic and higher-order data sets of two or more dimensions. 

Closure Models Many closure models have been proposed. A few of the more important ones are introduced here. Many employ the Boussinesq approximation, simphfied here for incompressible flow, which treats the Reynolds stresses as analogous to viscous stresses, introducing a scalar quantity called the turbulent or eddy viscosity... 

Expanding /g around the global equilibrium solution /eq at u = 0 in available scalar products using the vectors cg and u, we have, formally, in the homogeneous fluid approximation (Vm = 0),... 

Table 5.1 Effect of relativity on Hartree-Eock orbital energies (in eV) for the neutral Hg and Fe atoms. Scalar relativistic effects were treated with the DKH2 approximation...

Here, /3 and / are constants known as the Bohr magneton and nuclear magneton, respectively g and gn are the electron and nuclear g factors a is the hyperfine coupling constant H is the external magnetic field while I and S are the nuclear and electron spin operators. The electronic g factor and the hyperfine constant are actually tensors, but for the hydrogen atom they may be treated, to a good approximation, as scalar quantities. 

In the previous section the g value was considered as a scalar quantity, which was indeed a good approximation since the unpaired electron on the hydrogen atom occupies a spherically symmetric s orbital. If the unpaired electron exhibits p or d character the electron possesses both spin and orbital angular momentum. As a result the spin is not quantized exactly along the direction of the external field and the g value becomes a tensor... 

Here scalar order parameter, has the interpretation of a normalized difference between the oil and water concentrations go is the strength of surfactant and /o is the parameter describing the stability of the microemulsion and is proportional to the chemical potential of the surfactant. The constant go is solely responsible for the creation of internal surfaces in the model. The microemulsion or the lamellar phase forms only when go is negative. The function/(<))) is the bulk free energy and describes the coexistence of the pure water phase (4> = —1), pure oil phase (4> = 1), and microemulsion (< ) = 0), provided that/o = 0 (in the mean-held approximation). One can easily calculate the correlation function (4>(r)(0)) — (4>(r) (4>(0)) in various bulk homogeneous phases. In the microemulsion this function oscillates, indicating local correlations between water-rich and oil-rich domains. In the pure water or oil phases it should decay monotonically to zero. This does occur, provided that g2 > 4 /TT/o — go- Because of the < ), —<(> (oil-water) symmetry of the model, the interface between the oil-rich and water-rich domains is given by... 

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