Continuous representation

Numerical integration methods discussed so far provide information about the solution of the ODE at certain discrete points. The location of these points is determined by the step size control. Often however, the numerical solution is needed at other points, so-called output points. This situation occurs due to 

In the first case the location of the output points is often not known explicitly and is determined iteratively by solving an implicit relation. This will be the topic of Ch. 6. 

In all cases one is interested in obtaining a functional description of the numerical approximation to the solution rather than in its values only at discrete points. 

The goal is to construct and to evaluate a continuous representation without additional evaluations of the right hand side function /. The construction of a continuous representation is straight forward for integration methods based on a polynomial representation of the solution or its derivative, like Adams or BDF multistep methods or Runge-Kutta methods based on collocation. 

If Xn 0), 0 1 is used for output or post processing purposes only, it suffices 

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The NRT formalism will be used to describe the interacting species along the entire reaction coordinate. Such a continuous representation allows the TS complex to be related both to asymptotic reactant and product species and to other equilibrium bonding motifs (e.g., 3c/4e hypervalent bonding Section 3.5). A TS complex can thereby be visualized as intermediate between two distinct chemical bonding arrangements, emphasizing the relationship between supramolecular complexation and partial chemical reaction. 

Note that, unlike with the continuous representation (A.2), the discrete representation used in the SR model requires an explicit model for the scalar dissipation rate ea. 

There are a variety of formalisms that allow for a mapping of a discrete quantum system onto a continuous analog (for reviews see Refs. 218 and 219). The most prominent examples are Schwinger s theory of angular momentum [98] and the Holstein-Primakoff transformation [97], both of which allow a continuous representation of spin degrees of freedom. To discuss these two theories, we consider a spin DoF that is described by the spin operators Si,S2,Si with commutation relations... 

Let us begin our study of the problem by reformulating it in the continuous representation, with definitions of variables that have enjoyed some popularity ... 

Difficulties in solving the deconvolution problem are revealed if we examine it in the continuous representation given by Eq. (86). Suppose that the solution for o(x) is to some degree uncertain, and that it may be written as the sum of a desired solution o(x ) and a spurious part 0(x ) ... 

In this expression n and m are no longer restricted to integer values. As shown in Fig. 18.2 for n = 4 and n = 6, the continuous representation of the Bernoulli coefficients is surprisingly good even for small n values it becomes even better if n is large. 

The principles and applications of FEM are described extensively in the literature (e.g., 23-26). FEM is a numerical approximation to continuum problems that provides an approximate, piecewise, continuous representation of the unknown field variables (e.g., pressures, velocities). 

Before a detailed presentation of the ab initio dynamics simulations, first the fundamental difference between atomic and molecular adsorption on the one hand and dissociative adsorption on the other hand has to be addressed. Then I will briefly discuss the question whether quantum or classical methods are appropriate for the simulation of the adsorption dynamics. This section will be followed by a short introduction into the determination of potential energy surfaces from first principles and their continuous representation by some analytical or numerical interpolation schemes. Then the dissociative adsorption and associative desorption of hydrogen at metal and semiconductor surfaces and the molecular trapping of oxygen on platinum will be discussed in some detail. 

It is important to note that DFT total-energy calculations do not provide a continuous potential energy surface, as one might naively assume from the inspection of Fig. 2. In fact, the elbow plots shown are based on a series of 50-100 DFT calculations with varying center of mass and H-H distance. The continuous representation is just a result of a contour plot routine that interpolates between the actually calculated energies. 

For any dynamical simulation, a continuous representation of the PES is mandatory since the potential and the gradients are needed for arbitrary configurations. One can in fact perform ab initio molecular dynamics simulations in which the forces necessary to integrate the classical equations of motion are determined in each step by an electronic structure calculations. There have been few examples for such an approach [35-37], However, in spite of the fact that electronic structure calculations can nowadays be performed very efficiently, still there is a significant numerical effort associated with ab initio calculations. This effort is so large that in the ab initio dynamics simulations addressing molecular adsorption and desorption at surfaces the number of calculated trajectories has been well below 100, a number that is much too low to extract any reliable reaction probabilities. 

It is usual in laminar mixing simulations to represent the flow using tracer trajectories. The computation of such flow trajectories in a coaxial mixer is more complex than in traditional stirred tank modelling due to the intrinsic unsteady nature of the problem (evolving topology, flow field known at a discrete number of time steps in a Lagrangian frame of reference). Since the flow solution is periodic, a node-by-node interpolation using a fast Fourier transform of the velocity field has been used, which allowed a time continuous representation of the flow to be obtained. In other words, the velocity at node i was approximated... 

Note that in (7.3) time is a continuous variable, while the position n is discrete. We may go into a continuous representation also in position space by substituting... 

Note that even though in (7.5) we use a continuous representation of position and time, the nature of our physical problem implies that Ax and AZ are finite, of the order of the mean free path and the mean free time, respectively. 

Historic Examples of Multiscale Modeling The treatment of discrete and continuous representations of the vibrating string in Mathematical Thought from Ancient to Modern Times by Morris Kline, Oxford University Press, New York New York 1972, is enlightening. Here it is evident that both the continuous and discrete representations had features that recommended them as the basis for further study. 

The one-field model corresponding to the standard continuous representation of polymers is the Landau-Ginzburg model which we shall now review. 

The intensity interference pattern illustrated by Fig. 6.8 for bound-bound transitions is very similar to the Beutler-Fano lineshape for bound-free transitions discussed in Sections 7.9 and 8.9. The 101,000-106,000 cm-1 region of Fig. 6.8 is a band-by-band rather than a continuous representation of a Fano profile with q < 0 [see Fig. 7.26 and compare Eq. (7.9.6) to Eq. (6.3.17)]. 

Those of the former school—namely Turnbull and Bagley," LeFevre," Kratky," Gordon et al., Hoare," Hiwatari, and Frenkel and McTague —either imply or explicitly assert that the virial series could be a continuous representation of the supercooled fluid and the amorphous solid, with its first singularity at the zero-temperature point of the ground-state glass. This would require that the essentially Arrhenius behavior of... 

A final refinement, which is not described here in detail due to space limitations, involves the derivation of an approximate continuous representation of the CF3 F excitation energy distribution from the data hsted in Table XIII (47,48,49). This is possible based on accurate fractional decomposition yields as input data due to the serial nature of the cascade sequence shown in Reactions 81-84. The continuous distribution corresponding to Figure 3 is shown in Figure 5. 

The perturbation theory we consider is for the static reactivity pertaining to a perturbation of the reactor from a reference critical state. We use the general form of the transport equations with spelled-out notations and a continuous representation of the phase-space variables (r, E, 1). The subscript 0 is omitted from the parameters corresponding to the critical reactor. Instead, a bar denotes perturbed parameters. We take 7=1. 

A dynamical system C is a triplet U,G,a], where f/is an algebra C, G is a local compact group, and a is a strongly continuous representation of G in the group of the automorphisms of U, so that for each element g e G, is an automorphisms of U, with satisfying the following relations ... 

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