Differential equations, “stiff

In Sec. 5,7, we showed that the stability of the numerical solution of differential equations depends on the value of hX, and that A together with the stability boundary of the method determine the step size of integration. In the case of the linear differential equation 

A is the eigenvalue of that equation, and it remains a constant throughout the integration. The nonlinear differential equation 

The value of A is no longer a constant but varies in magnitude at each step of the integration. This analysis can be extended to a set of simultaneous nonlinear differentiai equations  

The eigenvalues A I i = 1,2. of the Jacobian matrix are the detennining factors in the stability analysis of the numerical solution. The step size of integration is determined by the stability boundary of the method and the maximum eigenvalue. 

When the eigenvalues of the Jacobian matrix of the differential equations are all of the same order of magnitude, no unusual problems arise in the integration of the set. However, when the maximum eigenvalue is several orders of magnitude larger than the minimum eigenvalue, the equations are said to be stiff. The stiffness ratio (SR) of such a set is defined as 

Recently attention has been focused in the chemical engineering literature on stiff systems. These are sets of differential equations that contain a mixture of very fast dynamic equations and very slow dynamic 

However, often the real problem is not with the numerical algorithm but with the engineer developing the equation set. If one is interested in the slower dynamic parts of the problem, a quasi-steady-state assumption should be made for the fast parts of the problem. On the other hand, if one is interested in the fast parts of the problem, the value of the slower parts essentially remains constant over these very short time periods. Therefore, stiff systems of equations should not arise in most properly formulated simulations that use order-of-magnitude scaling in model formation. 

MATLAB has two new stiff integration routines. These are ode 15s and ode23s. The routine odelSs is a variable order (up to order 5) and a variable step size program that is based upon the Klopfenstein modification of classical backward difference formulas called numerical differential formulas (Klopfenstein, 1971). Standard backward difference formulas are also available as an option. In order to determine optimum step size and speed convergence of the implicit corrector formulas, the method depends upon the Jacobian, J, of the derivative function / in 

A low order option is available as ode23. It is based upon a modified Rosenbrock single step formula similar to the Runge-Kutta method except it includes a Jacobian term. The method is therefore considered implicit since it requires the solution of systems of linear equations (Steihaug and Wolfbrandt, 1979). 

It should be noted that these two programs ode 15s and ode23s can handle sets of equations with a Mass Matrix, Af 

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With the introduction of Gear s algorithm (25) for integration of stiff differential equations, the complete set of continuity equations describing the evolution of radical and molecular species can be solved even with a personal computer. Many models incorporating radical reactions have been pubHshed. 

Spin trap, 102 Statistical kinetics, 76 Steady-state approximation, 77-82 Stiff differential equations, 114 Stoichiometric equations, 12 Stopped-flow method, 253-255 Substrate titration, 140 Success fraction approach, 79 Swain-Scott equation, 230-231... 

If the equation solver permits it, information can also be provided about the Jacobean of dealing with stiff differential equations. The Jacobean is of the form... 

LES/FDF-approach. An In situ Adaptive Tabulation (ISAT) technique (due to Pope) was used to greatly reduce (by a factor of 5) the CPU time needed to solve the set of stiff differential equations describing the fast LDPE kinetics. Fig. 17 shows some of the results of interest the occurrence of hot spots in the tubular LDPE reactor provided with some feed pipe through which the initiator (peroxide) is supplied. The 2004-simulations were carried out on 34 CPU s (3 GHz) with 34 GB shared memory, but still required 34 h per macroflow time scale they served as a demo of the method. The 2006-simulations then demonstrated the impact of installing mixing promoters and of varying the inlet temperature of the initiator added. 

Equations 1 and 2 can be solved numerically using an algorithm which handles stiff differential equations (28). Two sets of boundary conditions are required. For 0reduced catalyst is exposed to NO, the inlet gas composition is given by... 

Stiction, in mercury thermometers, 24 465 Stiff differential equation, 25 285 Stiffness loss, in fatigue, 16 187-188 of fibers, 11 181, 182 Stiffness values, of paper, 18-101 Stilbene(s), 25 181... 

All of the pertinent variables are now differential functions of the time parameter. These are stiff equations, however, that can be solved using an appropriate stiff differential equation solver. 

Recently, a number of very efficient and transportable software packages have become available for the solution of stiff differential equations involved in detailed chemical kinetic modeling (see, for example, Hindmarsh, 1980 Petzold, 1982 Caracatsios and Steward, 1985). Consequently, the actual solution of equations no longer limits the modeling process. Instead, the limiting factor today is the availability of reliable and fundamentally based chemical reaction mechanisms. 

The first two sections of Chapter 5 give a practical introduction to dynamic models and their numerical solution. In addition to some classical methods, an efficient procedure is presented for solving systems of stiff differential equations frequently encountered in chemistry and biology. Sensitivity analysis of dynamic models and their reduction based on quasy-steady-state approximation are discussed. The second central problem of this chapter is estimating parameters in ordinary differential equations. An efficient short-cut method designed specifically for PC s is presented and applied to parameter estimation, numerical deconvolution and input determination. Application examples concern enzyme kinetics and pharmacokinetic compartmental modelling. 

M72 Solution of stiff differential equations semi-implicit Runge-Kutta method with backsteps Rosenbrock-Gottwa1d-Wanner 7200 7416... 

Existence and uniqueness of the particular solution of (5.1) for an initial value y° can be shown under very mild assumptions. For example, it is sufficient to assume that the function f is differentiable and its derivatives are bounded. Except for a few simple equations, however, the general solution cannot be obtained by analytical methods and we must seek numerical alternatives. Starting with the known point (tD,y°), all numerical methods generate a sequence (tj y1), (t2,y2),. .., (t. y1), approximating the points of the particular solution through (tQ,y°). The choice of the method is large and we shall be content to outline a few popular types. One of them will deal with stiff differential equations that are very difficult to solve by classical methods. Related topics we discuss are sensitivity analysis and quasi steady state approximation. 

REN SOLUTION OF STIFF DIFFERENTIAL EQUATIONS t 7204 REN SENI INPLICIT-RUNGE KUTTA NETHOD NITH BACKSTEPS ... 

The quasi steady state approximation is a powerful method of transforming systems of very stiff differential equations into non-stiff problems. It is the most important, although somewhat contradictive technique in chemical kinetics. Before a general discussion we present an example where the approximation certainly applies. 

B.A. Gottwald and G. Wanner, A reliable Rosenbrock-integrator for stiff differential equations, Computing, 26 (1981) 335-357. 

Initial value problems, abbreviated by the acronym IVP, can be solved quite easily, since for these problems all initial conditions are specified at only one interval endpoint for the variable. More precisely, for IVPs the value of the dependent variable(s) are given for one specific value of the independent variable such as the initial condition at one location or at one time. Simple numerical integration techniques generally suffice to solve IVPs. This is so nowadays even for stiff differential equations, since good stiff DE solvers are widely available in software form and in MATLAB. 

So-called stiff differential equation models are particularly challenging to solve. Stiff models have dynamic behavior that encompasses a wide range of time scales. An example would be fast kinetics combined with long fluid-residence times in a chemical reactor. Gear s method is perhaps the most commonly used technique for solving these types of problems. 

As an alternative to the simultaneous solution of stiff differential equations through an implicit technique a method is described here which approximates the solution by successive computations of the corresponding finite difference equations. The successive nature of this method essentially decouples the K(N + 1)... 

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. 

Stiff differential equations Differential equations with widely varying rate constants. Like neural networks, their solution depends upon careful selection of step sizes. 

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. 

This is the characteristic equation for the eigenvalues posed by the Orr-Sommerfeld equation that also can be viewed as the dispersion relation of the problem. So the task at hand is to obtain a combination of a and u> for a given Re, such that the solution of OSE satisfies (2.4.8). The stiffness of OSE causes the numerical solution to lose the linear independence of different solution components corresponding to the different fundamental solutions. This is the source of parasitic error growth of any stiff differential equation. To remove this problem in a straight forward manner, one can use the Compound Matrix Method (CMM). 

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