Adaptive step size control

To control the step size adaptively we need an estimate of the local truncation error. With the Runge - Kutta methods a good idea is to take each step twice, using formulas of different order, and judge the error from the deviation between the two predictions. Selecting the coefficients in (5.20) to give the same a j and d values in the two formulas at least for some of the internal function evaluations reduces the overhead in calculation. For example, 6 function evaluations are required with an appropriate pair of fourth-order and fifth-order formulas (ref. 5). 

In the predictor - corrector methods the magnitude of the first correction is an immediate error estimate with no additional cost. 

From the actual step size h. ., error estimate Eest and the desired error bound Ecjes a new step size hnew can be selected according to 

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. 

Stiffness occures in a problem if there are two or more very different time scales on which the dependent variables are changing. Since at least one component of the solution is fast , a small step size must be selected. There is, however, also a slow variable, and the time interval of interest is large, requiring to perform a large number of small steps. Such models are common in many areas, e.g., in chemical reaction kinetics, and solving stiff equations is a challenging problem of scientific computing. 

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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. 

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... 

The numerical integration of (9.74) was carried out using the fourth-order Runge-Kutta procedure with adaptive step-size control as described in detail by Press et al. (1992). The parameters in the continuum description of the energy were chosen to be = 0.03 J/m, /3i = 15 J/m, tq = IJ/m, Cm = —0.01 and (Ss = 2.86 J/m. It is assumed that the elastic modulus and the Poisson s ratio are lO N/m and 0.3, respectively, for both the film and substrate materials. For the compressively strained film, the surface energy of the film (sketched schematically in Figure 8.26) attains a minimum when 6 = 0.12, which implies that the sidewalls of the stepped mounds would evolve naturally toward this angle. 

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]). 

Given this estimate, s +, and a specified target tolerance for the discretization error, the adaptive method will decrease the step size to comply with the tolerance. If the accuracy is met unecessarily well, the step size can be increased to reduce the computational effort. Furthermore, the step size can be controlled or bounded by a specified minimum step size, hmm, and maximum step size, hmax, to improve stability. 

The present paper steps into this gap. In order to emphasize ideas rather than technicalities, the more complicated PDE situation is replaced here, for the time being, by the much simpler ODE situation. In Section 2 below, the splitting technique of Maas and Pope is revisited in mathematical terms of ODEs and associated DAEs. As implementation the linearly-implicit Euler discretization [4] is exemplified. In Section 3, a cheap estimation technique for the introduced QSSA error is analytically derived and its implementation discussed. This estimation technique permits the desired adaptive control of the QSSA error also dynamically. Finally, in Section 4, the thus developed dynamic dimension reduction method for ODE models is illustrated by three moderate size, but nevertheless quite challenging examples from chemical reaction kinetics. The positive effect of the new dimension monitor on the robustness and efficiency of the numerical simulation is well documented. The transfer of the herein presented techniques to the PDE situation will be published in a forthcoming paper. 

The SMAD method relies on trapping metal atoms in a cold matrix formed by a condensed liquid, followed by a growth step that occurs when the matrix is warmed to room temperature. By controlling factors such as solvent polarity and the rate of heating, the size of the nanocrystals can be tuned. The reaction is carried out in a vacuum chamber filled with liquid vapors. The walls of the chamber are cooled to liquid nitrogen temperature and the metal atoms are introduced by thermally evaporating a metal chunk. The atoms upon evaporation get embedded in the solvent matrix condensed on the walls. Klabunde and coworkers [458] have adapted this method to S3mthesize Au nanocrystals... 

Tank capacity must also be considered when designing a winery. High-capacity tanks are of course economical, but tank size should not be exaggerated and should be adapted to the winery (50-350 hi). It is difficnlt to control the various steps in winemaking in vats containing over 350 hi. Tanks of limited capacity permit snperior batch selection and skin extraction dne to increased skin contact. The tank shonld be filled before the start of fermentation and for this reason the filling time should not exceed 12 hours. 

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