Step-size adjustment

The downward step size adjustment loop from lines 166 to 185 exits under two possible conditions - either the relative error meets the desired specification (see line 176) or the step size is at flic specified minimum value (see line 175). After... 

Listing 11.27. Code changes needed for using adaptive step size adjustment with the p-n junction problem. See Listing 11.26 for other lines of code. 

The Gear Algorithm [15], based on the Adams formulas, adjusts both the order and mesh size to produce the desired local truncation error. BuUrsch and Sloer method [16, 22] is capable of producing accurate solutions using step sizes that arc much smaller than conventional methods. Packaged Fortran subroutines for both methods are available. 

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. 

Miller, M.A. Amon, L.M. Reinhardt, W.P., Should one adjust the maximum step size in a Metropolis Monte Carlo simulation Chem. Phys. Lett. 2000, 331, 278-284... 

ABS(dely/y) compares it with the specified value MAXINC, and adjusts the step size if adjustment is needed biginc - 0... 

Difficulty 3 can be ameliorated by using (properly) finite difference approximation as substitutes for derivatives. To overcome difficulty 4, two classes of methods exist to modify the pure Newton s method so that it is guaranteed to converge to a local minimum from an arbitrary starting point. The first of these, called trust region methods, minimize the quadratic approximation, Equation (6.10), within an elliptical region, whose size is adjusted so that the objective improves at each iteration see Section 6.3.2. The second class, line search methods, modifies the pure Newton s method in two ways (1) instead of taking a step size of one, a line search is used and (2) if the Hessian matrix H(x ) is not positive-definite, it is replaced by a positive-definite matrix that is close to H(x ). This is motivated by the easily verified fact that, if H(x ) is positive-definite, the Newton direction... 

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. 

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 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. 

Once all of the conditions were determined and parameters chosen, the equations were solved by an implicit Euler method. The program was written with a self adjusting step size and analytic Jacobian to reduce error and run time. 

We used NVT MD to sample the contents of the simulation cell, and an NpT MC algorithm to vary its shape and volume. The latter moves were carried out within a rigid-molecule framework as described previously [76], using the atomic positions at the end of the preceding flexible molecule NVT MD segment. In practice, 1 ps of NVT MD simulation was followed by a sequence of 100 NpT MC steps. The Monte Carlo step size for a given thermodynamic state was adjusted to yield an acceptance probability of 40-50%. 

The state variable profiles of the model are assumed to be continuous and are obtained by integration of the DAEs over the entire length of the time. Also efficient integration methods (as available in the literature) are based on variable step size methods and not on fixed step size method where the step sizes are dynamically adjusted depending on the accuracy of the integration required. Therefore, the discrete values of the state variables are obtained using linear interpolation... 

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