The problem of stiffness

It is obviously beneficial for the simulation program to run as fast as possible, both in terms of cost and in terms of convenience for the control engineer who has to interact with it. But a major problem arising with process plants is the wide variety of time constants inherent in them. If the integration timestep 

All models of realistic physical systems will possess a range of time constants, and hence a degree of stiffness. A system will not be seen as stiff if 5 10, but a system with 5 100 will certainly be regarded as stiff. The boundary between what is and what is not a stiff system lies somewhere in between, perhaps with S = 30. 

The time constant is a linear concept, which derives from the solution of a linear differential equation such as that used to model valve 1 in Section 2.2  

The solution to this equation from a starting condition of a i = 0 at f = 0 is  

The time constant, ri, determines how rapidly xi approaches xj. For example, after a step input in xj, the difference between x and xj will be less than 5% after a time period of 3 time constants has elapsed. 

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With systems of this type, it is often found that some of the participating species are present in low concentrations only. If these species are very reactive (i.e. transient intermediates), the stationary state approximation may be applied. In making this approximation, it is assumed that the concentrations of the transient species remain constant. Thus, the kinetic scheme is simplified significantly and it may be possible to avoid the problem of stiffness . The validity of such assumptions must be examined carefully. As a simple example, consider the reaction scheme (17) for the case where Cb = 0. In applying the stationary state approximation to B, we have... 

Recently, several approaches have been proposed to overcome the disparity of time scales for certain classes of problems. In order to overcome the problem of stiffness caused by rapid, partial equilibrated reactions in a living free-radical polymerization system, a hybrid analytical-KMC method was suggested (He... 

Note that in (2.4.11), all the variables have the same exponential rate of growth or attenuation- a special feature of CMM where the problem of stiffness is removed by the specific choice of new variables. In integrating (2.4.10), one can start off with the values given by (2.4.11), with variables normalized with respect to one of them. Let us normalize every variable with respect to j/i, so that the initial conditions for solving Eqns. (2.4.10) are then. 

Values of delayed neutron parameters and the problem of stiffness... 

To avoid the problem of stiffness [ 134] in the vicinity of the singular point z = -1, we accurately fit all the terms in Equation (7.29) within a small initial interval z e [-1, -1 + e] by finite-order polynomial functions of (z -I-1) and then explicitly find a solution in the infinite order expansion... 

The question of stiffness then depends on the solution at the current time. Consequently nonhuear problems can be stiff during one time period and not stiff during another. While the chemical engineer may not actually calculate the eigenvalues, it is useful to know that they determine the stabihty and accuracy of the numerical scheme and the step size used. 

The smallest eigenvalue is independent of Ax (it is DK /L ) so that the ratio of largest to smallest eigenvalue is proportional to /Ax. Thus, the problem becomes stiff as /Ax approaches zero (Ref. 106). 

The error in Runge-Kutta calculations depends on h, the step size. In systems of differential equations that are said to be stiff, the value of h must be quite small to attain acceptable accuracy. This slows the calculation intolerably. Stiffness in a set of differential equations arises in general when the time constants vary widely in magnitude for different steps. The complications of stiffness for problems in chemical kinetics were first recognized by Curtiss and Hirschfelder.27... 

Nearly anytime chemical reactions enter a problem, the notion of stiffness comes to the fore. Stiffness only has practical meaning in the context of numerical solutions. Because... 

The modified Newton iteration, and the reason that damping is effective, can be explained in physical terms. Chemical-kinetics problems often have an enormous range of characteristic scales—this is the source of stiffness, as discussed earlier. These problems are also highly nonlinear. 

The objective of this problem is to explore the performance of stiff and nonstiff user-oriented initial-value-problem software. Acquire the Fortran source code and the documentation for Vode from www.netlib.org. The VODE package enables the user to select either stiff or nonstiff methods. 

Solve the nominal problem for an initial condition of A = 0.9 and B = 0.1. Explain the short-time and long-time behavior of the solution in the context of stiffness. 

The towed sled is the most common form of test and many such apparatus have been devised. Whilst simple in principle, there are practical problems in that the sled will tend to tilt if it is not towed on the plane where the surfaces meet, and if the means of applying the force is a wire or cord, the lack of stiffness can cause slip-stick. 

A survey over the area of stiff-chain polyelectrolytes has been given. Such rod-like polyelectrolytes can be realized by use of the poly(p-phenylene) backbone [9-13]. The PPP-polyelectrolytes present stable systems that can be studied under a wide variety of conditions. Moreover, electric birefringence demonstrates that these macroions form molecularly disperse solution in water [49]. The rod-like conformation of these macroions allows the direct comparison with the predictions of the Poisson-Boltzmann cell model [27-30] which has been shown to be a rather good approximation for monovalent counterions but which becomes an increasingly poor approximation for higher valent counterions [29]. Here it was shown in Sect. 2.2 that the basic problem of the PB model, namely the neglect of correlations, can be remedied in a systematic fashion. 

Warner [176] has given a comprehensive discussion of the principal approaches to the solution of stiff differential equations, including a hundred references among the most pertinent books, papers and application packages directed at simulating kinetic models. Emphasis has been put not only on numerical and software problems such as robustness, improving the linear equation solvers, using sparse matrix techniques, etc., but also on the availability of a chemical compiler, i.e. a powerful interface between kineticist and computer. 

Finite difference — Finite difference is an iterative numerical procedure that has been used to quantify current-voltage-time relationships for numerous electrochemical systems whose analyses have resisted analytic solution [i]. There are two generic classes of finite difference analysis 1. explicit finite difference (EFD), where a new set of parameters at t + At is computed based on the known values of the relevant parameters at t and 2. implicit finite difference (IFD), where a new set of parameters at t + At is computed based on the known values of the relevant parameters at t and on the yet-to-be-determined values at t + At. EFD is simple to encode and adequate for the solution of many problems of interest. IFD is somewhat more complicated to encode but the resulting codes are dramatically more efficient and more accurate - IFD is particularly applicable to the solution of stiff problems which involve a wide dynamic range of space scales and/or time scales. 

This article deals with some topics of the statistical physics of liquid-crystalline phase in the solutions of stiff chain macromolecules. These topics include the problem of the phase diagram for the liquid-crystalline transition in die solutions of completely stiff macromolecules (rigid rods) conditions of formation of the liquid-crystalline phase in the solutions ofsemiflexible macromolecules possibility of the intramolecular liquid-crystalline ordering in semiflexible macromolecules structure of intramolecular liquid crystals and dependence of die properties of the liquid-crystalline phase on the microstructure of the polymer chain. 

Q = [a +iaRe —c)Y/ . For boundary layer instability problems, i e —> 00 and then Q >> laj. This is the source of stiffness that makes obtaining the numerical solution of (2.3.21) a daunting task. This causes the fundamental solutions of the Orr-Sommerfeld equation to vary by different orders of magnitude near and far away from the wall. This type of behaviour makes the governing equation a stiff differential equation that suffers from the growth of parasitic error, while numerically solving it. 

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