Experiments numerical

Let us consider testing the computed first order corrections on a one dimensional example. Let M = P = and define the biased double-well potential as 

We shall consider computing the average of a specific observable (q,p), and estimating the error due to discretization using the correction functions given in (7.25), where we choose the observable 

The numerical experiment at each timestep is averaged from 16 independent experiments, each running over a time interval of 10. The results match our 

The first order correction term for the [BOA scheme is given by the formula 

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Numerical Experiment We now present numerical results for non-zero Vi and V2- In particular, we take... 

These various techniques were recently applied to molecular simulations [11, 20]. Both of these articles used the rotation matrix formulation, together with either the explicit reduction-based integrator or the SHAKE method to preserve orthogonality directly. In numerical experiments with realistic model problems, both of these symplectic schemes were shown to exhibit vastly superior long term stability and accuracy (measured in terms of energy error) compared to quaternionic schemes. 

In this article, we briefly describe these symplectic methods, citing recent articles for most of the details of derivation and implementation. We compare the various algorithms in terms of theoretical and implementation aspects, as well as in simple numerical experiments. 

This paper is meant as a contribution to systematize the quantum-classical modeling of molecular dynamics. Hence, we are interested in an extended theoretical understanding of the models rather than to further contribute to the bunch of numerical experiments which have been performed on certain models by applying them to particular molecular systems. Thus, we will carefully review the assumptions under which our models are known to approximate the full quantum dynamical (QD) evolution of the system. This knowledge... 

D. Okunbor, Canonical methods for Hamiltonian systems Numerical experiments , Physica D, 60, 314-322, 1992. 

In 1875 Alexander M Zaitsev of the University of Kazan (Russia) set forth a gen erahzation describing the regioselectivity of p eliminations Zaitsev s rule summarizes the results of numerous experiments m which alkene mixtures were produced by p elim matron In its original form Zaitsev s rule stated that the alkene formed in greatest amount is the one that corresponds to removal of the hydrogen from the f3 carbon hav mg the fewest hydrogens... 

Numerous experiments on rodents, as well as dogs and monkeys, with dosage levels up to 43 g of MSG per kilogram of body weight have failed to show any link between dietary use of MSG and brain damage. In the case of dogs and monkeys, even experiments involving injection of MSG have not shown any effects on the brain. 

There have been numerous experiments using this technique, mostly in shallow waters, in Europe and N. America. With the exception of the Norfolk Broads Restoration programme, there have been few attempts to resolve eutrophication by bio-manipulation in UK lakes. 

No fixed rules can be given it is up to the experience of the engineer to judge the necessary steps. Good advice, however, is to define a model with only the flow feature of concern, to test several levels of simplification on this model, and to decide from these numerical experiments the level of simplification for the full model under investigation. 

In addition, Strehlow et al. (1979) performed numerical experiments on accelerating flames. Their conclusions may be summarized as follows ... 

Numerical experiments by Langton [langQOa] show once again that when Xm is plotted against A, there is a region for small A for which Xm is essentially zero, a critical A at which Xm jumps to some moderate value followed by a slow decay as... 

The solution of equation 5 for flux provides the kinetics we desire. Numerous experiments with the hexane slurry system have led to the development of an expression for k that is partly based on theory and partly on an empirical constant in the denominator ... 

The reaction between a Lewis acid R3M and a Lewis base ER3, usually resulting in the formation of a Lewis acid-base adduct R3M—ER3, is of fundamental interest in main group chemistry. Numerous experiments, in particular reactions of alane and gallane MH3 with amines and phosphines ER3, have been performed [14]. Several general coordination modes, as summarized in Fig. 2, have been identified by X-ray diffraction. 

The blue color of 83 has been observed in numerous experiments. For example, a brilliant blue color occurs if a potassium thiocyanate melt is heated to temperatures above 300 °C [132] or if eutectic melts of LiCl-KCl (containing some sulfide) are in contact with elemental sulfur [132, 133], if aqueous sodium tetrasulfide is heated to temperatures above 100 °C [134], if alkali polysulfides are dissolved in boiling ethanol or in polar aprotic solvents (see above), or if borate glasses are doped with elemental sulfur [132]. In most of these cases mixtures of much 83 and little 82 will have been present demonstrating the ubiquitous nature of these radicals [12]. 

A volume correction is necessary, as can be seen from a numerical experiment similar to the preceding one If the increase in volume from VI to VI + V2 is ignored, a bias of about 7% is produced. 

The results from the model were compared semi-quantitatively with experimental measurements to validate the model. Because of both the extremely long time constants in the system and the variations in the ambient conditions in the laboratory where the extruder was placed, it was not possible to rigorously test the model against experimental data. To conserve demands of time and materials for experiments on the extruder, numerical experiments were used to provide data for developing an optimal control system. The goal of the numerical experiments was to develop a reduced-order model suitable for optimizing the control system. 

From numerical experiments in the range from 200 to 220 C, the following constants were determined ... 

The numerical experiment started at a steady-state value of 200 C for both temperature nodes with an output of 16.89% for both heaters output number 1 was then stepped to 19.00%. If both outputs had been stepped to 19%, then both nodes would have gone to 220 C. The temperature of node 5 does not go as high, and the temperature of node 55 goes too high. In the reduced order model, the time constant x represents the effect of radial heat conduction, while the time constant X2 represents the effect of axial heat conduction. SimuSolv estimates these two parameters of the dynamic model as ... 

A greater gain in accuracy in connection with the temperature wave depends significantly on how well we calculate the coefficients a (v). In the case where k = k u is a power function of temperature, numerical experiments showed that formula (38) is useless and formula (36) is much more flexible than (37), so there is some reason to be concerned about this. Further comparison of schemes (34) and (35) should cause some difficulties. Both schemes are absolutely stable and have the same error of approximation 0 r + h ). The scheme a) is linear with respect to the value of the function on the layer and so the value y7+i on every new layer... 

No wishing to cover the computational procedures of MATM once again, we fill in the table on the basis of numerical experiments for e = 10 . ... 

The stages of numerical experiments for theoretical investigations of physical problems were explained above. Greatest attention is being paid to a new technology for research with rather complicated mathematical models. 

Any numerical experiment is not a one-time calculation by standard formulas. First and foremost, it is the computation of a number of possibilities for various mathematical models. For instance, it is required to find the optimal conditions for a chemical process, that is, the conditions under which the reaction is completed most rapidly. A solution of this problem depends on a number of parameters (for instance, temperature, pressure, composition of the reacting mixture, etc.). In order to find the optimal workable conditions, it is necessary to carry out computations for different values of those parameters, thereby exhausting all possibilities. Of course, some situations exist in which an algorithm is to be used only several times or even once. 

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