Stiff system

We described earlier (cf. Sec. 4.1.5) stiffness as a property of a method, when applied to a certain differential equation with a given tolerance. 

the term stiff differential equation is used to indicate that special methods are used for numerically solving them. These methods are called stiff integrators and are characterized by A-stability or at least i4(a)-stability. They are always implicit and require a corrector iteration based on Newton s method. For example BDF methods or some implicit Runge-Kutta methods, like the Radau method are stiff integrators in that sense. 

Though there is no rigorous definition of a stiff differential equation we will give here some characteristics and demonstrate some phenomena. 

Stiff differential equations are characterized by a solution composed by fast and slow modes. If a fast mode is excited, this leads to a steep transient to the solution component given by the slow mode, see Fig. 4.9. 

When using a numerical method with a bounded stability region and a too large step size, a fast mode causes an instability in the numerical solution, so the step size must be restricted in this case, often drastically. 

Equilibrium is established only when the intensive variables corrected by the transformation factors are equal. We may not conclude what will happen if the system is not in equilibrium, whether it will move to an equilibrium state or rather away. We may think that a system is more far from equilibrium as the difference of intensive variables is more large. In the example above, we may take a, the width of the lamella w, the mass m, and g as constants. In this case, equilibrium is at crw = gm. Under laboratory conditions, we may easily change the width of the lamella w and the hanging mass m, but the other parameters are difficult to change. In this case, equilibrium is established only in one singular point. If only one parameter does not fit, equilibrium is not established. On the other hand, in the case of the soap bubble, inside there is a compressible gas. We have the well-known Kelvin s equation (for a double surface) 

Massieu, M.F. Sur les fonctions des divers fluides. Compt. Rendus Acad. Set. 69, 858-862 (1869) 

Poincare, L. The New Physics and its Evolution. The International Scientific Series. D. Appleton and Company, New York (1909). Translation of La physique modeme, son evolution, Project Gutenberg eBook, [eBook 15207] 

Barkai, E., Silbey, R.J. Fractional Kramers equation. J. Phys. Chem. B 104, 3866- 

van den Broeck, C., Garcia, A. Rectification of thermal fluctuations in ideal gases. 

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Because of the inherently greater susceptibility of expansion bellows to failure from unexpected corrosion, failure of guides to control joint movements, etc., it is advisable to examine critically their design choice in comparison with a stiff system. 

Ordinaiy differential Eqs. (13-149) to (13-151) for rates of change of hquid-phase mole fractious are uouhuear because the coefficients of Xi j change with time. Therefore, numerical methods of integration with respect to time must be enmloyed. Furthermore, the equations may be difficult to integrate rapidly and accurately because they may constitute a so-called stiff system as considered by Gear Numerical Initial Value Problems in Ordinaiy Differential Equations, Prentice Hall, Englewood Cliffs, N.J., 1971). The choice of time... 

Fixed step, 2nd-order, implicit method for stiff systems (ALGO = 3). 

As we have already pointed out in this chapter for systems described by algebraic equations, the introduction of the reduced sensitivity coefficients results in a reduction of cond(A). Therefore, the use of the reduced sensitivity coefficients should also be beneficial to non-stiff systems. 

The most common numerical problem, as shown by some of the simulation examples, is that of equation stiffness. This is manifested by the need to use shorter and shorter integration step lengths, with the result that the solution proceeds more and more slowly and may come to a complete halt. Such behaviour is exhibited by systems having combinations of very fast and very slow processes. Stiff systems can also be thought of as consisting of differential equations, having large differences in the process time constants. Sometimes, the... 

In summary, in the equilibrium-chemistry limit, the computational problem associated with turbulent reacting flows is greatly simplified by employing the presumed mixture-fraction PDF method. Indeed, because the chemical source term usually leads to a stiff system of ODEs (see (5.151)) that are solved off-line, the equilibrium-chemistry limit significantly reduces the computational load needed to solve a turbulent-reacting-flow problem. In a CFD code, a second-order transport model for inert scalars such as those discussed in Chapter 3 is utilized to find ( ) and and the equifibrium com-... 

The braced frame will be designed using static design process based on the capacity of supported members. Bracing provides a stiff system which responds to pressure without-absorbing much energy. 

You may wonder why we would ever be satisfied with anything less than a very accurate integration. The ODEs that make up the mathematical models of most practical chemical engineering systems usually represent a mixture of fast dynamics and slow dynamics. For example, in a distillation column the liquid flow or hydraulic dynamic response occurs fairly rapidly, of the order of a few seconds per tray. The composition dynamics, the rate of change of hquid mole fractions on the trays, are usually much slower—minutes or even hours for columns with many trays. Systems with this mixture of fast and slow ODEs are called stiff systems. 

The main advantage of the implicit algorithms is that they do not become numerically unstable. Very large step sizes can be taken without having to worry about the instability problems that plague the explicit methods. Thus, the implicit methods are very useful for stiff systems. 

Burrage, K., and Petzold, L. R., On order reduction for Runge-Kutta methods applied to differential/algebraic systems and to stiff systems of ODEs," Lawrence Livermore National Laboratory, UCR-98046 preprint (1988). 

In a stiff system, no variables are constrained, but the total potential energy of the system, which will be denoted by V Q), has a sharp minimum (i.e., an/-dimensional valley) along the constraint surface. To describe slight deviations from the constraint surface, we may Taylor expand V Q) to harmonic order around its value on the constraint surface, which will be denoted by U(q). This yields... 

We now extend the preceding analysis to the case of a stiff system, by using an extended local-equilibrium hypothesis to remove both momenta and hard coordinates from the problem, and thus obtain a diffusion equation for the distribution of soft coordinates alone. 

We first treat a stiff system as a generic unconstrained system. We consider a joint probability distribution T (g) for all 3N coordinates of a stiff system, soft and hard, given by... 

To construct a closed set of equations for the evolution of /((7) in a stiff system, we need an expression for the average solvent force that appears in... 

Eq. (2.119). A diffusion equation of the form given in Section IV is recovered if and only if we identify (Fa )f as a hydrodynamic drag force, and (as for the rigid system) assume that it may be described by a generalized Stokes equation of the form given in Eq. (2.74), where U is defined for a stiff system by Eq. (2.106). 

This equation is equivalent to the result given by Ottinger for the Ito drift velocity of an arbitrary rigid system, generalized here by the use of an unspecified value of W so as to also apply to stiff systems. This drift velocity appears as the drift coefficient (i.e., the prefactor of dt) in the Ito SDE given in Eq. (29) of Ref. 11 andEq. (5.61) of Ref. 12. 

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