Stiff integration

At the other extreme of Distefano s sample problems, for the largest initial charge, the maximum-stiffness ratio is of the order of 1500, which is considered to be a relatively large value. In this case, more than 10,000 time steps are required to distih 90 percent of the initial change, and the problem is better handled by a stiff integrator. 

TOLUENE/O-XYLENE SEPARATION AT 90 MM HC RIGOROUS VAPOR-HYDRAULIC MODEL (VAPOR RATES calculated from pressure drop through TRAYS) USING LSODE IMPLICIT STIFF INTEGRATOR PROGRAM ASSUMPTIONS ... 

Try to solve the Oregonator model using a non - stiff integrator as the module M70. Comment an the step size needed for a reasonable accuracy. 

As seen from the output of Example 5.4A, the solution of the system (5.50-53) is far from easy even for the stiff integrator M72. In the following we solve the same problem applying the quasi steady state approx imation. 

Note further that inside fluidbed.m we use the stiff integrator odel5s instead of one of our previous favorites ode23 or ode45, since the latter two integrators take an inordinate... 

How do we choose a MATLAB IVP solver, and how do we decide whether to use a stiff integrator such as odel5s or a standard nonstiff one ... 

Thus due to potential stiffness, integrating a BVP such as the one in equation (5.37) in the positive uj direction from 0 to 1 may not be wise in all cases. In fact, backward integration is much more stable for our simplified model since in backward integration, the eigenvalues switch signs and then the problem is no longer stiff according to the definition. 

The set of four ordinary differential equations (7.64) to (7.67) for the dynamical system are quite sensitive numerically. Extreme care should be exercised in order to obtain reliable results. We advise our students to experiment with the standard IVP integrators ode... in MATLAB as we have done previously in the book. In particular, the stiff integrator odel5s should be tried if ode45 turns out to converge too slowly and the system is thus found to be stiff by numerical experimentation. 

Stiff IVPs are best solved by the integrators of the MATLAB ode.. . suite of functions that end in the letter s. In practical terms, one need not construct a linearization to check for stiffness, but rather compare the run times for ordinary integrators (without an s in their MATLAB name) and for stiff integrators (with an s ). If the ordinary ones take too long, the IVP problem is most likely stiff and a MATLAB integrator with a name ending in s should be used to solve the problem more successfully. 

Depending on the numerical techniques available for integration of the model equations, model reformulation or simplified version of the original model has always been the first step. In the Sixties and Seventies simplified models as sets of ordinary differential equations (ODEs) were developed. Explicit Euler method or explicit Runge-Kutta method (Huckaba and Danly, 1960 Domenech and Enjalbert, 1981 Coward, 1967 Robinson, 1969, 1970 etc) were used to integrate such model equations. The ODE models ignored column holdup and therefore non-stiff integration techniques were suitable for those models. 

Tackling stiffness in process simulations the properties of a stiff integration algorithm... 

We will now illustrate the way that equation (2.70) could be solved as part of a stiff integration package. The solution relies partly on using the New-ton-Raphson technique for solving nonlinear simultaneous equations, the principles of which will now be explained. We may describe a system of n nonlinear. 

A common method of solution is the Newton-Raphson method, already described in connection with a stiff integration algorithm in Section 2.7, equations (2.71) to (2.74). The equations above are in the form... 

Table 2 shows the mass balance of the main components of the process in the fluid and solid phase. The whole set of equations is solved coupling the Orthogonal Collocation Method to discretize the radial ordinates with the Method of Lines to integrated the system of equations by a stiff integrator. This work is only a theoretical study, as a predictive tool, and all presented data were simulated. 

The Newmark s p method (linear stiffness integration method) is also applied to take account of the non-linear part of the whole structure. If the linear stiffiiess is much larger than the non-linear stiffness, the integration method is unconditionally stable. In this work, the equations were transformed into an incremental form. It was confirmed that even when the incremental form of the equations is applied, the condition where the linear stiffness is larger than the non-linear tangent stiffness ensures that the integration method gives a stable solution. 

It is, however, well-known to be quite challenging to any numerical stiff integrator due to its oscillatory behavior any errors introduced at some integration step will not die out furtheron, since the system is not dissipative. It is... 

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... 

More complex oscillations have been found when the full TWC microkinetic model (Eqs. 1-31 in Table 1) has been used in the computations, cf. Fig. 4. The complex spatiotemporal pattern of oxidation intermediate C2H2 (Fig. 4, right) illustrates that the oscillations result from the composition of two periodic processes with different time constants. For another set of parameters the coexistence of doubly periodic oscillations with stable and apparently unstable steady states has been found (cf. Fig. 5). Even if LSODE stiff integrator (Hindmarsh, 1983) has been succesfully employed in the solution of approx. 10 ODEs, in some cases the unstable steady state has been stabilised by the implicit integrator, particularly when the default value for maximum time-step (/imax) has been used (cf. Fig. 5 right and Fig. 3 bottom). Hence it is necessary to give care to the control of the step size used, otherwise false conclusions on the stability of steady states can be reached. 

Often, 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. 

When using a non-stiff integration method as MATLAB s Runge-Kutta method ODE45, much more effort is necessary to obtain a solution as can be seen from the table beside. 

AM97] Arnold M. and Murua A. (1997) Non-stiff integrators for differential-algebraic systems of index 2. Annals of Numerical Mathematics, submitted for publication. 

For columns where the plate dynamics are significantly faster than the reboiler dynamics (due to very small plate holdups and/or wide boiling components), the stiff integrator often fails to find a solution. The solution to this problem is to split the system into two levels (a) the reboiler, where the dynamics are slower, can be represented by differential equations (Equations 4.12-4.13), and (b) the rest of the column can be assumed to be in the quasi-steady state. Thus, the composition changes in the condenser and accumulator (dx /dt), the composition changes on plates (dx j /dt), and the enthalpy changes in the condenser and on plates (Stlo... 

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