Computational instability

Several research groups have concentrated on the problem of computational instability. The failure of c culations has often been attributed to approximation errors, which refer to the inability of the numerical method to fit the set of governing equations. So far, numerical studies have revealed that the usueQ computational instability, occurring with simple but unrealistic models, was overcome to a great extent by the use of new integral models, which have enabled... 

Computational instability The exponential growth of the numerical solution of the differential equation. 

Phillips NA (1959) An example of non-linear computational instability. In The Atmosphere and Sea in Motion, pp 501-504, Rockefeller Inst Press, New York, and Oxford University Press... 

The mole fractions at the end of the time increment (equivalent to the distillate increment) are calculated by numerical integration. The magnitude of the time increment is determined by stability and truncation considerations. The batch distillation model contains tray holdups with time constants much smaller than the reboiler time constant. These conditions ( stiff systems ) can cause computational instability unless very small time increments are used. The penalty is excessive computing time and the likelihood of incurring truncation errors. Distefano (1968) provides values for the maximum time increment size consistent with stability for a number of integration schemes. The same time increment is used to determine the incremental distillate rate for the flrst step. 

Figure 7.13. Computed C2 reaction path for dimerization of cyclopropene.The numbering of the atoms follows that of Chemical Abstracts for tricyclo[3.1.0.0 ]hexane. The slight computational instability near r4s = 1.7A is due to a discontinuity in the limited Cl procedure used.

The existence of truncation errors in finite difference approximations to differential equations is discussed in numerical analysis texts with respect to round-off error and computational instabilities (Roache, 1972 Richtmyer and Morton, 1957), but Lantz (1971) was among the first to address the form of the truncation error as it related to diffusion. Lantz considered a linear, convective, parabolic equation similar to 9u/9t + U 9u/9x = e S u/Sx and differenced it in several ways. He showed that the effective diffusion coefficient was not 8, as one might have suggested analytically, but 8 + 0(Ax, At) (so that the actual diffusion term appearing in computed solutions is the modified coefficient times c2u/9x2) where the 0(Ax,At) truncation errors, being functions of u(x,t), are comparable in magnitude to 8. Because this artificial diffusion necessarily differs from the actual physical model, one would expect that the entropy conditions characteristic of the computed results could likely be fictitious. 

Although its origins date back much earlier. Computational Chemistry evolved into a distinctly discernible discipline three decades ago from the conjunction of many developments. Effective computers were a prerequisite. While the very first electronic calculating machines had been employed for quantum chemistry applications, the limitations of the first commercial computers (instability, vacuum tubes with all their drawbacks, single user, little or no memory, punch cards or paper tape, cumbersome operating systems) restricted the interest to a few dedicated theoreticians. 

Since the drop volume method involves creation of surface, it is frequently used as a dynamic technique to study adsorption processes occurring over intervals of seconds to minutes. A commercial instrument delivers computer-controlled drops over intervals from 0.5 sec to several hours [38, 39]. Accurate determination of the surface tension is limited to drop times of a second or greater due to hydrodynamic instabilities on the liquid bridge between the detaching and residing drops [40],... 

A second case to be considered is that of mixtures witli a small size ratio, <0.2. For a long time it was believed tliat such mixtures would not show any instability in tire fluid phase, but such an instability was predicted by Biben and Flansen [109]. This can be understood to be as a result of depletion interactions, exerted on the large spheres by tire small spheres (see section C2.6.4.3). Experimentally, such mixtures were indeed found to display an instability [110]. The gas-liquid transition does, however, seem to be metastable witli respect to tire fluid-crystal transition [111, 112]. This was confinned by computer simulations [113]. 

Errors in advection may completely overshadow diffusion. The amplification of random errors with each succeeding step causes numerical instability (or distortion). Higher-order differencing techniques are used to avoid this instability, but they may result in sharp gradients, which may cause negative concentrations to appear in the computations. Many of the numerical instability (distortion) problems can be overcome with a second-moment scheme (9) which advects the moments of the distributions instead of the pollutants alone. Six numerical techniques were investigated (10), including the second-moment scheme three were found that limited numerical distortion the second-moment, the cubic spline, and the chapeau function. 

It must be pointed out that the heterofuUerenes discussed above are not available today, and may never be available owing to synthetic limitations or unexpected instability not predicted in the above-mentioned theoretical studies. In comparison to carbon bucky balls, the chemistry of heterofuUerenes might have more important implications. Development of molecular engines and computers, derivatization for drug delivery, and applications in material science might be new scientific areas involving these interesting molecules. 

This model does not say anything about the mechanism of triple-helix formation, because even in the case of an AON mechanism, nucleation may take place at many positions of the chains and may lead to products the chains of which are staggered. The AON model is based on the assumption that these products are too instable to exist in measurable concentration. As already mentioned, Weidner and Engel142 succeeded in proving by relaxation measurements of al CB2 that the kinetics of in vitro triple-helix formation is governed by more than one relaxation time. This rules out an AON mechanism, but the fitting to the experimentally found equilibrium transition curves nevertheless showed good accommodation and AH° computed from these curves could be confirmed by calorimetric measurement. 

The objectives of this presentation are to discuss the general behavior of non isothermal chain-addition polymerizations and copolymerizations and to propose dimensionless criteria for estimating non isothermal reactor performance, in particular thermal runaway and instability, and its effect upon polymer properties. Most of the results presented are based upon work (i"8), both theoretical and experimental, conducted in the author s laboratories at Stevens Institute of Technology. Analytical methods include a Semenov-type theoretical approach (1,2,9) as well as computer simulations similar to those used by Barkelew LS) ... 

The lumped parameter model of Example 13.9 takes no account of hydrodynamics and predicts stable operation in regions where the velocity profile is elongated to the point of instability. It also overestimates conversion in the stable regions. The next example illustrates the computations that are needed... 

Sung, C.J., Makino, A., and Law, C.K., On stretch-affected pulsating instability in rich hydrogen/air flames Asymptotic analysis and computation. Combust. Flame, 128, 422, 2002. 

The above two examples were chosen so as to point out the similarity between a physical experiment and a simple numerical experiment (Initial Value Problem). In both cases, after the initial transients die out, we can only observe attractors (i.e. stable solutions). In both of the above examples however, a simple observation of the attractors does not provide information about the nature of the instabilities involved, or even about the nature of the observed solution. In both of these examples it is necessary to compute unstable solutions and their stable and/or unstable manifolds in order to track and analyze the hidden structure, and its implications for the observable system dynamics. 

Within esqjlicit schemes the computational effort to obtain the solution at the new time step is very small the main effort lies in a multiplication of the old solution vector with the coeflicient matrix. In contrast, implicit schemes require the solution of an algebraic system of equations to obtain the new solution vector. However, the major disadvantage of explicit schemes is their instability [84]. The term stability is defined via the behavior of the numerical solution for t —> . A numerical method is regarded as stable if the approximate solution remains bounded for t —> oo, given that the exact solution is also bounded. Explicit time-step schemes tend to become unstable when the time step size exceeds a certain value (an example of a stability limit for PDE solvers is the von-Neumann criterion [85]). In contrast, implicit methods are usually stable. 

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