Space discretization

Where several units of the same type are mounted within a space, discrete frequencies will be amplified and beat notes will be apparent. Special treatment is usually called for, in the way of indirect air paths and mass-loaded panels [10, 19, 56, 60]. 

By representing the operator containing the potential energy in position state space and the one containing the kinetic energy in momentum space, one obtains the following phase space discretized path integral representation ... 

The Fourier series, which has a discrete spectrum but periodic spatial function, is actually a special case of the Fourier transform. (Note that an equally spaced discrete spectrum necessarily implies a periodic function having a finite period given by the wavelength of the lowest frequency.) See Bracewell (1978) to see how the explicit form of the Fourier series may be obtained from the Fourier transform. Taking discrete, equally spaced... 

In order to perform space discretization, the domain over which the governing equations apply is filled with a predetermined mesh or grid. The mesh is made up of nodes (i.e. grid points) and/or elements at which the physical quantities (i.e. unknowns) are evaluated. Neighbouring points are used to calculate derivatives. Mesh generation is a very complex task for applied problems and many different approaches to it have been developed and are currently under study [74-77], First of all, computational grids are classified as structured or unstructured meshes, even if each of these classes comprises a broad fist of meshing techniques. 

In the case of dynamic (unsteady) problems, even after the space discretization, we still have to solve a set of ordinary differential equations in time. Therefore, the second step is to discretize the temporal continuum. This is usually done by a finite difference approximation with the same properties of a FDM in space. Depending on the instant in which the information is taken, the time-discretization leads to ... 

Unfortunately, the result shown in Figure 2 corresponds to time-shared (i.e., essentially simultaneous) excitation/detectlon, so that the (discrete) frequencies sampled by the detector are the same as those initially specified in synthesis of the time-domain transmitter signal. However, FT/ICR is more easily conducted with temporally separated excitation and detection periods. In practical terms, the result is that for FT/ICR, we need to know the excitation magnitude spectrum at all frequencies, not just those (equally-spaced) discrete frequencies that defined the desired excitation spectrum. 

This PWE was used in [18] to obtain the numerical results. For the numerical implementation the B-spline approximation [21] was chosen that represents actually the refined version of the space discretization approach. In Table 1 the convergence of the PWE approach with the multicommutator expansion is presented for the lowest-order SE correction for the ground state of hydrogenlike ions with Z = 10. The minimal set of parameters for the numerical spline calcuations was chosen to be the number of grid points N = 20, the number of splines k = 9. This minimal set allowed to keep a controlled inaccuracy below 10%. What is most important for the further generalization of the PWE approach to the second-order SESE calculation is that with Zmax = 3 the inaccuracy is already below 10% (see Table 1). The same picture holds with even higher accuracy for larger Z values. The direct renormalization approach is not necessarily connected with the PWE. In [19] this approach in the form of the multicommutator expansion (Eq. (16)) was employed in combination with the Taylor expansion in powers of (Ea — En>)r 12 The numerical procedure with the use of B-splines and 3 terms of Taylor series yielded an accuracy comparable with the PWE-expansion with Zmax = 3. 

We now return to the loop-after-loop SESE calculations in [11]. The first two terms of the potential expansion Eq. (2), ZP and OP terms were evaluated in momentum space. For this purpose the Fourier transform was performed for the bound state wave functions n) in coordinate space. The latter were evaluated by the space discretization method. The MP term was calculated entirely in coordinate space. 

Let us briefly discuss the relationship between approaches which use basis sets and thus have a discrete single-particle spectrum and those which employ the Hartree-Fock hamiltonian, which has a continuous spectrum, directly. Consider an atom enclosed in a box of radius R, much greater than the atomic dimension. This replaces the continuous spectrum by a set of closely spaced discrete levels. The relationship between the matrix Hartree-Fock problem, which arises when basis sets of discrete functions are utilized, and the Hartree-Fock problem can be seen by letting the dimensions of the box increase to infinity. Calculations which use discrete basis sets are thus capable, in principle, of yielding exact expectation values of the hamiltonian and other operators. In using a discrete basis set, we replace integrals over the continuum which arise in the evaluation of expectation values by summations. The use of a discrete basis set may thus be regarded as a quadrature scheme. 

The time method of lines (continuous-space discrete-time) technique is a hybrid computer method for solving partial differential equations. However, in its standard form, the method gives poor results when calculating transient responses for hyperbolic equations. Modifications to the technique, such as the method of decomposition (12), the method of directional differences (13), and the method of characteristics (14) have been used to correct this problem on a hybrid computer. To make a comparison with the distance method of lines and the method of characteristics results, the technique was used by us in its standard form on a digital computer. 

The method of characteristics, the distance method of lines (continuous-time discrete-space), and the time method of lines (continuous-space discrete-time) were used to solve the solids stream partial differential equations. Numerical stiffness was not considered a problem for the method of characteristics and time method of lines calculations. For the distance method of lines, a possible numerical stiffness problem was solved by using a simple sifting procedure. A variable-step fifth-order Runge-Kutta-Fehlberg method was used to integrate the differential equations for both the solids and the gas streams. 

Hilbert space of a finite-dimensional normed vector space discrete description. However, there are a number of cases where the extension to an infinitedimensional Hilbert space cannot be dealt with in too cavalier a fashion, and these will be discussed as the need arises at this stage, we simply discuss a few essential points about the mathematical formalism connected with a continuous description. 

In this section, a survey of the basic elements of the finite volume method, as applied to single phase flows, is provided [141, 201, 202, 49, 158]. The numerical issues considered are the approximations of surface and volume integrals, time discretizations, and space discretization of diffusive and convective (or advective) terms. 

Fractional step methods have become quite popular. There are many variations of them, due to a vast choice of approaches to time and space discretizations, but they are all based on the principles described above. To... 

The spatial structure of the MMGEP model is determined by the databases available. Spatial inhomogeneity is provided for by the various forms of space discretization. The basic type of spatial discretization of the Earth surface is a uniform geographic grid with arbitrary latitude and longitude steps. 

Here Waa ( = L, R, C) denote the (still infinite) Hamiltonians of the left and right lead and the finite size Hamiltonian of the central device, respectively. The latter is chosen so large that direct interactions between the leads can be neglected. The partitioning in Eq. (2.6) may be realized by space discretization or by a representation of the Hamiltonian in a suitable localized basis. Inserting Eq. (2.6) into the time-dependent KS equations Eq. (2.3), leads to a reduced equation of motion for the wavefunction IV c) in the central device region ... 

This is mainly due to the development of numerical methods (B-spline approach [62], space discretization [63]) that allow summations to be performed over the complete Dirac spectrum for arbitrary spherically symmetric potentials. 

To parametrize the paths, we use q uniformly spaced discrete points for each of the two real-time paths and r points for the imaginary-time path (see Fig. 1). Hence there are M = 2q- - r points in total. The time discretizations are... 

Consider the integral equation li[/(r)] = 0 for a function /(r) defined in the 3D space, discretized on a grid of k points. The solution is thus represented by a vector f r-k), and the integral equation reduces to a set of k nonlinear equations zeroing the values of the residual li[/(r)] at the points rjt, that is zeroing the residual vector... 

The collinear A+BC dissociative collision can be treated in a straightforward manner,using the Time Dependent WavePacket (TDWP) method. The reason is that the dissociative continuum of the BC molecule is handled automatically within the space discretization scheme of the grid. As the basic method has already been described in detail elsewhere ,it will be only outlined here,emphasizing the technical points and some new features which lead to a significant reduction in computation time. 

For every equally spaced discrete frequency of interest/, (>0), [Pg.451]

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