Adaptive step size

The correction to the relaxing density matrix can be obtained without coupling it to the differential equations for the Hamiltonian equations, and therefore does not require solving coupled equations for slow and fast functions. This procedure has been successfully applied to several collisional phenomena involving both one and several active electrons, where a single TDHF state was suitable, and was observed to show excellent numerical behavior. A simple and yet useful procedure employs the first order correction F (f) = A (f) and an adaptive step size for the quadrature and propagation. The density matrix is then approximated in each interval by... 

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... 

Learning of Type-2 Fuzzy Logic Systems by Simulated Annealing with Adaptive Step Size... 

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. 

Forgetting for the moment accuracy and adaptive step-size selection, consider how the eqnation is used given the state, we evaluate the time derivative and then, nsing mathematics (in particular, Taylor series) and smoothness to create a local hnear model of the process in time, we make a prediction of the state at the next time step. A numerical integration code will ping a subroutine with the current state as input and will obtain as output the time derivative at this state. The code will then process this value and use local Taylor series to make a prediction of the next state (the next value of c at which to call the subroutine evaluating the function j). 

Why not use higher-order methods beyond the fourth order One reason is that the Runge-Kutta methods with higher order use more function evaluations, which means that the number of computations increases. The objective of numerical solutions is to obtain a solution with sufficient accuracy, not an exact solution. Consequently, it is often not necessary to use higher-order methods because the improvement in accuracy is offset by the increase in computational effort. A better strategy for increasing accuracy is instead to use adaptive step size methods. 

Adaptive step size methods and error controi... 

Explain what is meant by adaptive step size methods, and why they are nsed. 

One approach to an adaptive step size is simply to first assume some best guess at a good step distribution and then calculate the solution and obtain an estimated error as has been done in the previous section. Based upon the resulting global error a new step size distribution could be developed and the calculation could be repeated, hopefully until a given accuracy had been achieved. This would proba-... 

Code segment for odeivsQ an adaptive step size integration algo-... 

Listing 10.18. Example of use of adaptive step size algorithm for solving three differential equations. 

Figure 10.26. Comparison of solution errors and solution values for the stiff differential equation example using adaptive step size algorithm. Solution from Listing 10.18.

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