Numerical integration step size

The computer model consists of the numerical integration of a set of differential equations which conceptualizes the high-pressure polyethylene reactor. A Runge-Kutta technique is used for integration with the use of an automatically adjusted integration step size. The equations used for the computer model are shown in Appendix A. 

Most commonly used ordinary differential equation (ODE) solvers provide options of several different integration techniques. Most solvers also automatically vary the integration step size during the simulation to allow the best trade-off between accuracy and solution time, based on user-specified numerical tolerances. There is no single best integration technique—different methods work better for various problems. 

To assure sufficient accuracy of the numerical integration, we repeatedly halved the integration step size until no significant difference (0.1% ) in solutions occurred. For the fourth-order Runge-Kutta method this required a step size of. 04 sec (43 steps). An upper bound of the total error introduced by the numerical integration procedure can be obtained for the Runge-Kutta method (29). At a step size of 0.04 sec, the error estimate calculated is 0.004% (14). 

This involves the knowledge of the order of the discontinuity. Numerical approaches to determine this order work only reliable for orders 0 and 1 [G084]. We therefore suggest to restart the integration method for safety reasons. When restarting a multistep method, much time is spent for regaining the appropriate integration step size and order. The best one can do is to use the information available from the last time interval before the discontinuity in the case of the BDF method for... 

Figure C.l shows the Simulink diagram for Eq. C-1 (transport delay 1 = 5 for both models). To generate a transient response, the simulation menu is selected to allow parameters for the simulation to be specified (start time, stop time, integration routine, maximum integration step size). Numerical values of time t are entered into the input-output data set via a clock block. After the simulation has been completed, the resulting data can be plotted (see Fig. C.2), manipulated, and analyzed from the MATLAB command window.

R. D. Skeel and J. J. Biesiadecki, Symplectic integrations with variable step-size , Annals Numer. Math., 191-198, 1994. 

Step size is critical in all sim tilation s. fh is is th c iricrcm en t for in tc-grating th c equation s of motion. It uitim atcly deterrn in cs the accuracy of the numerical integration. For rn olecu les with high frequency motion, such as bond vibrations that involve hydrogens, use a small step size. 

Starting with an initial value of and knowing c t), Eq. (8-4) can be solved for c t + At). Once c t + At) is known, the solution process can be repeated to calciilate c t + 2At), and so on. This approach is called the Euler integration method while it is simple, it is not necessarily the best approach to numerically integrating nonlinear differential equations. To achieve accurate solutions with an Eiiler approach, one often needs to take small steps in time. At. A number of more sophisticated approaches are available that allow much larger step sizes to be taken but require additional calculations. One widely used approach is the fourth-order Bunge Kutta method, which involves the following calculations ... 

To obtain the results in Table 20.2, equations (xii) and (xiii) are solved numerically using E-Z Solve (file ex20-5.msp) and die initial conditions above for cA and cD. For N = 1, equation 13.4-2 is used for Ex(t) and is integrated to obtain W,(t), and similarly for N = 2 and equation 17.2-4. Note that most software for numerical integration cannot directly handle the negative step sizes required to solve the maximum-mixedness model... 

In contrast to our preferred standard mode in this book, we do not develop a Matlab function for the task of numerical integration of the differential equations pertinent to chemical kinetics. While it would be fairly easy to develop basic functions that work reliably and efficiently with most mechanisms, it was decided not to include such functions since Matlab, in its basic edition, supplies a good suite of fully fledged ODE solvers. ODE solvers play a very important role in many applications outside chemistry and thus high level routines are readily available. An important aspect for fast computation is the automatic adjustment of the step-size, depending on the required accuracy. Also, it is important to differentiate between stiff and non-stiff problems. Proper discussion of the difference between the two is clearly outside the scope of this book, however, we indicate the stiffness of problems in a series of examples discussed later. So, instead of developing our own ODE solver in Matlab, we will learn how to use the routines supplied by Matlab. This will be done in a quite extensive series of examples. 

The fourth order Runge-Kutta method is the workhorse for the numerical integration of ODEs. Elaborate routines with automatic step-size control are available in Matlab. 

This technique is quite simple and easy to visualize and program. It is not very rapid in converging to the correct solution, but it is rock-bottom stable (it won t blow up on you numerically). It works well in dynamic simulations because the step size can be adjusted to correspond approximately to the rate at which the variable is changing with time during the integration time step. 

With hmax = ma,x(xi+i — x/), the global error order of the classical Runge-Kutta method is of order 4, or 0(h/nax), provided that the solution function y of (1.13) is 5 times continuously differentiable. The global error order of a numerical integrator measures the maximal error committed in all approximations of the true solution y(xi) in the computed y values y. Thus if we use a constant step of size h = 10 3 for example and the classical Runge-Kutta method for an IVP that has a sufficiently often differentiable solution y, then our global error satisfies... 

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