Numerical Realization

The results presented here were obtained by applying a three-dimensional finite element method (FEM) to eq.(4). However this formalism is to complicated and thus not pedagogical to be used directly in a review text like the present one. We thus first introduce a FEM-realization of the one-dimensional Schrodinger problem[33]. 

The finite element method is employed to solve the eigenvalue equation 

The matrix elements H,mjn and Simjn are = 0 when i j. Therefore the matrices H and 5 are block-diagonal. 

Due to the block-diagonal form of the matrices the method is fast and numerically stable. 

We defined the elementary functions as functions, which are non-zero only inside a given element. On rectangular 3-dimensional elements this reads 

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A convenient and constructive approach to attain symplectic maps is given by the composition of symplectic maps, which yields again a symplectic map. For appropriate Hk, the splittings (6) and (7) are exactly of this form If the Hk are Hamiltonians with respect to the whole system, then the exp rLnk) define the phase flow generated by these Hk- Thus, the exp TL-Hk) are symplectic maps on the whole phase space and the compositions in (6) and (7) are symplectic maps, too. Moreover, in order to allow for a direct numerical realization, we have to find some Hk for which either exp(rL-Kfc) has an analytic solution or a given symplectic integrator. 

Summarizing, the number of the iterations required during the course of MATM in an arbitrary complex domain is close to the number of the iterations performed for the same Dirichlet problem in a minimal rectangle containing the domain G and numerical realizations confirm this statement. 

Because of the enormous range of difference approximations to an equation having similar asymptotic properties with respect to a grid step (the same order of accuracy or the number of necessary operations), their numerical realizations resulted in the appearance of different schemes for solving the basic problems in mathematical physics. 

H. A. Makse and J. Kurchan, Thermodynamic approach to dense granular matter A numerical realization of a decisive experiment. Nature 415, 614 (2002). 

The advantage of a graphical matrix lies in the fact that it allows for a great many possibilities of numerical realizations. In order to obtain a numerical form of a graphical matrix, one needs to select a graph invariant... 

Janezic, D., Lucic, B., Mdicevic, A., Nikolic, J., Trinajstic, N. and Vukicevic, D. (2007) Hosoya matrices as the numerical realization of graphical matrices and derived structural descriptors. Croat. Chem. Acta, 80, 271—276. 

Various methods have been described for the numerical realization of the respective model (Smith, 1965 Lasia, 1983). Since the appropriate boundary and initial conditions can also be employed with nonlinear reaction rates, the resulting system of parabolic differential equations... 

A Numerical realization of three-body quantum mechanics. 

To solve these problems the resultant method [10] or the Groebner basis method [91 can be used. It appears that the second method is more convenient as far as it has numerical realization in several computer algebraic systems. 

We note, that the Brooks-Corey function in pc = Pe and the van Genuchten function with n<2 for pc = 0 are not differentiable, which may cause trouble depending on the numerical realization. On the other hand, there appears to be no physical reason for having a discontinuity in the derivatives. 

The precise interpretation of the dynamics associated with the Monte Carlo procedure is that it is a numerical realization of a Markov process described by a master equation for the probabilty P(X, t) that a configuration X occurs at time t,... 

This suggests that to determine the path solutions in Eq. (4.88), j4(R, t) should be known beforehand. This is a nonlinear process. However, the Bohmian dynamics is known to be a very important reformulation of the Schrodinger equation with respect to the interpretation of quantum mechanics. Much study has been devoted to practical methods to numerically realize the Bohmian dynamics in the context of chemical dynamics (see [48f] and references therein). 

This section follows the above basic theory and method of path-branching representation of nonadiabatic dynamics with numerical examples. We also show the performance of the semiclassical Ehrenfest theory to compare with. What we present below is illustrative numerical realization rather than optimized practice for actual applications. 

We emphasis that the results presented in this subsection is totally independent of PSANB, which will be numerically realized below. 

The next stage considers carbon nanotubes with the refrigerant core, which are embedded in the nanostructure and positioned as heat outflows. In the numerical realization four nanotubes were considered to be embedded in the material on the comers of the area exposed to external heating. A nanotube was programmed as an infinitely thin source (outflow) of heat with constant power. Let us consider the temperature profiles in the observation areas. The observation points were taken similarly to the first stage of the study. Figure 16.11 presents the results of numerical simulation for the temperature distribution in points 1, 2 and 3 which spatial distribution was mentioned above. It is obvious that the heat outflows influence the temperature of nanostracture if earlier the design volume warmed up to the phase transition temperature, it can be seen that now the temperature on the axis of symmetry does not exceed the critical value. 

In a general case, displacement of oil by polymer and surfactant is effected by complex physico-chemical processes, when modeling and numerical realization of which there take place definite problems. For example, viscosity of injected solution depends on various factors, such as reservoir temperature, concentration of polymer/surfactant in solution and water salinity and etc. The model takes into account the following assumptions ... 

Common numerical realizations of the Eq. (1) used in Refs. [12, 15] enable us to calculate the kink structure and dynamics accurate enough to watch the kink pinning and passing through the impurities, structure and properties of the nonlinear waves excited. Yet much higher accuracy is required to study possible resonance effects. So N =10" points were used for approximating the function 0(x,t). Special control of the result errors is carried out. 

The thus specified Hamiltonians are then used in a dynamic Monte Carlo simulation (see Chapter 1) that is a numerical realization of the discrete time master equation... 

Solution of Eq. (8.5) allows for two representations according to which angle 4>n or ( )r is equal to zero. We have used the first case. A detailed description of this approach and its numerical realization is given in [27, 28, 36]. Here, we only state in brief the key ideas of the method. 

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