Step size control

If any component Ay is less than 0.001, we ignore that component in the following crude but practical tests. 

Michelsen used Eq. 7.125, which is a third order method, and performed two calculations at every time step. One calculation uses a step size of h and the other uses the step size of h/2. The error at the time is then defined 

Then for a tolerance of e, the step size is accepted when the following ratio is less than unity. 

Once the better estimate for y i is obtained (Eq. 7.137) the next step size is chosen as 

The exponent -1/4 comes from the fourth order method, resulting from the extrapolation. 

Numerical Solution of Ordinary Differential Equations Chapter 5 

A simple test for checking the step size is to do the calculations at each interval twice Once with the full step size, and then repeat the calculations over the same interval with a smaller step size, u.sually half that of the first one. If at the end of the interval, the difference between the predicted value of y by both approaches is less than the specified convergence criterion, the step size may be increased. Otherwise, a larger than acceptable difference between the two calculated y values suggests that the step size is large, and it should be shortened in order to achieve an acceptable truncation error. 

Another method of controlling the step size is to obtain an estimation of the truncation error at each interval. A good example of such an approach is the Runge-Kutta-Fehlberg method (.see Table 5.2), which provides the estimation of the local truncation error. This error estimate can be easily introduced into the computer program, and let the program automatically change the step size at each point until the desired accuracy is achieved. 

As mentioned before, the optimum number of application of corrector is two. Therefoi-e, in the case of using a predictor-corrector method, if the convergence is achieved before the second corrected value, the step size may be increased. On the other hand, if the convergence is not achieved after the second application of the corrector, the step size should be reduced. 

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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. 

For fast computation the determination of the best step-size (interval) is crucial steps that are too small result in correct concentrations at the expense of long computation times. On the other hand, steps that are too long save computation time but result in poor approximations. The best intervals lead to the fastest computation of concentration profiles within some pre-defined error limits. This of course requires knowledge about the required accuracy. The ideal step-size is not constant during the reaction and so needs to be adjusted continuously. If more complex mechanisms and thus systems of differential equations are to be integrated, adaptive step size control is absolutely essential. 

This function is called numerous times from the Matlab ODE solver. In the example it is the ode45 which is the standard Runge-Kutta algorithm. ode45 requires as parameters the file name of the inner function, ode autocat. m, the vector of initial concentrations, cO, the rate constants, k, and the total amount of time for which the reaction should be modelled (20 time units in the example). The solver returns the vector t at which the concentrations were calculated and the concentrations themselves, the matrix C. Note that due to the adaptive step size control, the concentrations are computed at times t which are not predefined. 

A way to check the accuracy of the solution is repeating the procedure with a smaller step size until the significant digits will be unchanged. More efficient methods of step size control will be discussed in Section 5.1.3. In this example the step size h = 0.05 hours has been proved appropriate and results in the following solution ... 

The most sophisticated differential equation solver considered in this book and discussed in the next section includes such step size control. In contrast to most integrators, however, it takes a full back step when facing a sudden increase of the local error. If the back step is not feasible, for example at start, then only the current step is repeated with the new step size. 

In addition to the solution, the number of accepted steps (ip) and the number of back steps or repeated steps (ir) are also printed to show how the step size control works. 

The step size control parameter R, initially set to a value of order unity, is adaptive, in the sense that it is decreased (or increased) at each iteration depending on how well (or how badly) the energy change actually brought about by the corrections accords with the value predicted from a second-order Taylor expansion in the corrections themselves. In extreme cases, the corrections are rejected and recomputed with an increased value of R. Otherwise, any updates to R apply from the next iteration. The precise set of rules used to control R may affect efficiency but is not critical to the success of the minimization procedure, as long as the rules provide the correct qualitative behaviour (e.g. see Refs. [22] and [18]). 

One particular class of ordinary differential equation solvers (ODE-solvers) handles stiff ODEs and these are widely known as stiff solvers. In our context, a system of ODEs sometimes becomes stiff if it comprises very fast and also very slow steps or relatively high and low concentrations. A typical example would be an oscillating reaction. Here, a highly sophisticated step-size control is required to achieve a reasonable compromise between accuracy and computation time. It is well outside the scope of this chapter to expand on the intricacies of modem numerical... 

The value of y" - z is an estimate of the error in y" and can be used in step-size control schemes. 

Often, the step size is chosen to increase the accuracy of the numerical solution. More importantly, the step size controls not only the error, but also the stability of the numerical scheme. The following problem is selected to illustrate the stability characteristics of the numerical method ... 

E = 0.1, 0.5, l.Oe above both the saddles. For the trajectory calculations, we used a fourth-order Runge-Kutta method with adaptive step-size control [61], and the total energies in their MD calculations were conserved within... 

Repeat Problem 7.6, but instead of a constant step size, use the step size control described in Section 7.10 for the Michelsen method. 

The resulting set of differential equations was solved numerically by means of an improved Euler method with variable step size control [9]. Parameter optimization was accomplished by a Mead Nelder simplex algorithm [10]. 

The adjustment of the step sizes of every parameter to the local topology of the quality function merits special attention both in the case of evolutionary algorithms and as far as other traditional direct optimization methods are concerned. If the step sizes selected are too high, the extreme value of a quality function is hit only by sheer coincidence. If, on the other extreme, the step sizes are too low, the speed of convergence falls and the computation time increases. Establishing the pace of mutation by a step size control determines the convergence of the process. 

One method of setting a step size control in the event of parameter problems as a default value is to multiply the rates of the parents with logarithmic, normally distributed random values, the standard deviations of which are normally distributed as well. Frequently, however, a better approach is an automatic adjustment of parameter rates by an appropriate adjustment to the local topology of the quality function. This, in turn, is achieved by the mutation selection process of the parameter sets, which, as additional control genes, contain their own step size for all parameters. In general, at the beginning of the optimization the rates should be set relatively high. The more closely an extreme value approaches, the more the rate should be reduced. 

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