Mathematically stiff equations

Next, we have to solve for the Yj s from the species continuity equations, Equation (32). Unfortunately, these equations cannot be integrated by a similar simple point iteration scheme as they are mathematically "stiff"16 and iterative approaches are unstable. To solve these simultaneous equations, we turn to a perturbation analysis developed by Newman17 where the equations are linearized about an initial guess, and the resulting linear equations are solved numerically. The solution is then used as the next guess, and the linear equations are resolved. The procedure is repeated until the solution no longer changes. 

The radical reaction schemes for thermal cracking mentioned in Chapter 1 have not been used so far in design. They lead to a set of continuity equations for the reacting components that are mathematically stiff in nature, because of the orders of magnitude of difference between the concentrations of molecular and radical species. Only recently have satisfactory numerical integration routines for sets of stiff differential equations been worked out (see Gear [17]). In addition, the rate parameters of radical reactions are frequently not known with sufficient precision, so far. The radical scheme has therefore been approximated by a set of reactions containing only molecular species. 

Pao YC, Nagendra GK, Padiyar R, Ritman EL (1980) Derivation of myocardial stiffness equation based on theory of laminated composites. J Biomech Eng 102 252-257 Peskin CS (1975) Mathematical aspects of heart physiology. Courant Institute of Mathematical Sciences, New York... 

In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. The main idea is that the equation includes some terms that can blow up the solution (i.e., the values of the dependent variable become infeasible or physically non-sense). 

E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, volume 14 of Springer Series in Computational Mathematics. Springer-Verlag, New York, New York, second edition, 1996. 

Mathematical models of the reaction system were developed which enabled prediction of the molecular weight distribution (MWD). Direct and indirect methods were used, but only distributions obtained from moments are described here. Due to the stiffness of the model equations an improved numerical integrator was developed, in order to solve the equations in a reasonable time scale. 

It is instructive to study a much simpler mathematical equation that exhibits the essential features of boundary-layer behavior. There is a certain analogy between stiffness in initial-value problems and boundary-layer behavior in steady boundary-value problems. Stiffness occurs when a system of differential equations represents coupled phenomena with vastly different characteristic time scales. In the case of boundary layers, the governing equations involve multiple physical phenomena that occur on vastly different length scales. Consider, for example, the following contrived second-order, linear, boundary-value problem ... 

From a mathematical point of view, we can see that Equation (5.10) is in a (nonstandard) singularly perturbed form. This suggests that the integrated processes under consideration will feature a dynamic behavior with at least two distinct time scales. Drawing on the developments in Chapters 2, 3, and 4, the following section demonstrates that these systems evolve in effect over three distinct time scales and proposes a method for deriving reduced-order, non-stiff models for the dynamics in each time scale. 

Egly et al. (1979), Cuille and Reklaitis (1986), Mujtaba (1989), Reuter et al. (1989), Albet et al. (1991), Basualdo and Ruiz (1995) and Wajge and Reklaitis (1999) considered the development of mathematical models to simulate BREAD processes. In most cases, the model was posed as a system of Differential and Algebraic Equations (DAEs) and a stiff solution method was employed for integration. 

The mathematical models of the reacting polydispersed particles usually have stiff ordinary differential equations. Stiffness arises from the effect of particle sizes on the thermal transients of the particles and from the strong temperature dependence of the reactions like combustion and devolatilization. The computation time for the numerical solution using commercially available stiff ODE solvers may take excessive time for some systems. A model that uses K discrete size cuts and N gas-solid reactions will have K(N + 1) differential equations. As an alternative to the numerical solution of these equations an iterative finite difference method was developed and tested on the pyrolysis model of polydispersed coal particles in a transport reactor. The resulting 160 differential equations were solved in less than 30 seconds on a CDC Cyber 73. This is compared to more than 10 hours on the same machine using a commercially available stiff solver which is based on Gear s method. 

Mathematical models that contain ordinary differential equations face an inherent computational difficulty associated with the stiffness of the equations. Stiffness of ordinary differential equations depends on the relative magnitudes of the response modes or the characteristic time constants of the system being modeled. In solid fuel conversion problems where particles of varying sizes are considered the differential equations for the thermal transients of the particles are usually stiff. Estab-... 

Over the past ten years the numerical simulation of the behavior of complex reaction systems has become a fairly routine procedure, and has been widely used in many areas of chemistry, [l] The most intensive application has been in environmental, atmospheric, and combustion science, where mechanisms often consisting of several hundred reactions are involved. Both deterministic (numerical solution of mass-action differential equations) and stochastic (Monte-Carlo) methods have been used. The former approach is by far the most popular, having been made possible by the development of efficient algorithms for the solution of the "stiff" ODE problem. Edelson has briefly reviewed these developments in a symposium volume which includes several papers on the mathematical techniques and their application. [2]... 

It is important to note that in using computer-aided models for batch distillation, the various assumptions of the model can have a significant impact on the accuracy of the results e.g., see the discussion of the effects of holdup above. Uncertainties in the physical and chemical parameters in the models can be addressed most effectively by a combination of sensitivity calculations using simulation tools, along with comparison to data. The mathematical treatment of stiffness in the model equations can also be very important, and there is often a substantial advantage in using simulation tools that take special account of this stiffness. (See the 7th edition of Perry s Chemical Engineers Handbook for a more detailed discussion of this aspect). 

Operation of a batch distillation is an rmsteady state process whose mathematical formulation is in terms of differential equations since the compositions in the stiff and of the holdups on individual trays change with time. This problem and methods of solution are treated at length in the literature, for instance, by Holland and Liapis (Computer Methods for Solving Dynamic Separation Problems, 1983, pp. 177-213). In the present section, a simplified analysis will be made of batch distillation of binary mixtures in colunms with negligible holdup on the trays. Two principal modes of operating batch distillation columns may be employed ... 

The mathematical model forms a system of coupled hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs). The model could be converted to a system of ordinary differential equations by discretizing the spatial derivatives (dx/dz) with backward difference formulae. Third order differential formulae could be used in the spatial discretization. The system of ODEs is solved with the backward difference method suitable for stiff differential equations. The ODE-solver is then connected to the parameter estimation software used in the estimation of the kinetic parameters. More details are given in Chapter 10. The comparison between experimental data and model simulations for N20/Ar step responses over RI1/AI2O3 (Figure 8.8) demonstrates how adequate the mechanistic model is. 

For stiff differential equations, the backward difference algorithm should be preferred to the Adams-Moulton method. The well-known code LSODE with different options was published in 1980 s by Flindmarsh for the solution of stiff differential equations with linear multistep methods. The code is very efficient, and different variations of it have been developed, for instance, a version for sparse systems (LSODEs). In the international mathematical and statistical library, the code of Hindmarsh is called IVPAG and DIVPAG. 

Thus in a few steps, equation 2.30 demonstrates mathematically the intuitively reasonable idea that a stiff chain contains fewer segments for a given chain length. Because the extension increases as the square root, this equation also shows why rubbers are composed of high polymer chains. 

These differences of reactivity are at the basis of the stiffness of the mathematical equations encountered in chemical kinetics and the consequent difficulties involved in obtaining a numerical solution. 

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