Region Runge-Kutta

X m, m = 0,..., 32, f = 1,2 with the help of a fourth order Runge-Kutta scheme (see, e.g., Milne (1970)). Every single one of the 62 resulting trajectories was followed over 200 cycles of the microwave field, and the values of I and 6 after every completion of a full cycle of the microwave field were plotted as dots in a 6,1) phase-space diagram. The result is shown in Fig. 6.5. Regular and chaotic regions are clearly visible. 

In ref 161 the authors have produced exponentially-fitted BDF-Runge-Kutta type formulas (of second-, third- and fourth-order). The good behaviour of the new produced methods for stilf problems is proved. The stability regions of the new proposed methods have been examined. It is proved that the plots of their absolute stability regions include the whole of the negative real axis. Finally the author propose several procedures to find the parameter of the new methods. 

The fourth-order explicit Runge-Kutta algorithm has a slightly better stability region than the Euler forward method. 

Phase separation was computer simulated using finite-difference in time and space Runge-Kutta and Monte Carlo with a Hamiltonian methods (Petschek and Metiu 1983 Meakin and Reich 1982 Meakin et al. 1983). Both methods were found equivalent, reproducing the observed pattern of phase separation in both NG and SD regions. The unity of the phase separation dynamics on both sides of the spinodal has been emphasized (Leibler 1980 Yemkhimovich 1982). 

In the same formula, h is the step size of the integration and n is the number of steps, i.e. is the approximation of the solution in the point x and Xn = XQ + n h and xq is the initial value point. Using the above methods and based on the theory of phase-lag and amplification error for the Runge-Kutta-Nystrom methods, the authors have derived two fourth algebraic order Runge-Kutta-Nystrom methods with phase-lag of order four and amplification error of order five. For both of methods the authors have obtained the stability regions. The efficiency of the produced methods is proved via numerical experiments. 

Since the problem is treated as an initial-value problem, one needs yo and y, before starting a two-step method. From the initial condition, yo = 0. The value yi is computed using the Runge-Kutta-Nystrom 12(10) method of Dormand et al. With these starting values we evaluate at Xi of the asymptotic region the phase shift d from the above relation. 

Figure 6. Average error r ons (%) for the modified Euler (a), third order Runge-Kutta (b), fourth order Runge-Kutta (c) and die Adams-Moulton (d) mediods. Areas indicated as having 1% error correspond to unstable regions.

Within the last region, a G (, oo), these asymptotic expansions describe the actual scalet solution (although large expansion orders may be required). In the tables, we indicate the scale value and expansion order (i.e. a N)) required for the asymptotic expansions to adequately approximate the Runge-Kutta expression for the scalet solution. 

Runge-Kutta-Nystrom method of Prince and Dormand. With these starting values we evaluate at Xi of the asymptotic region the phase shift 5i... 

Figure 4.6 Stability regions for the Runge-Kutta pair DOPRI4 and DOPRI5.The methods are stable inside the gray areas.

Figure 5.7 Stability region in the complex plane for Runge-Kutta methods of order 1 (explicit Euler), 2, 3, 4, and 5.

---