Slow time scales asymptotic solutions

Thus, the formulas (2.5), with C = 1, represent an appropriate continuation of the asymptotic solution (2.4) on the long time interval (on the slow time scale). Moreover, using the matching (2.6) one can both... 

Formulas (4.9), (4.10) provide the approximate solution on the slow time scale. From these asymptotics one can see that the second and third components tend to zero, whereas the first one tends to a nonzero constant y = (W/A) at infinity. If we use Taylor expansions of the left sides of Eqs. (4.9) and (4.10), the error estimate of the asymptotic behavior at infinity can easily be derived as follows ... 

In fact, the question under consideration is to find an asymptotic solution that describes the fast passage across the instant of explosive growth on the slow time scale 0 = v. From a formal point of view, the boundary asymptotic conditions are obtained in the same way, as was described in sections 4.2 and 4.3. Namely, the asymptotic expansions of the coefficients x, y",Zj(0) as are substituted into the series... 

In contrast, the asymptotic approach puts minimal strain on the computer but demands more of the modeller. The convergence of the computed solutions is usually easy to test with respect to spatial and temporal resolution, but situations exist where reducing the timestep can make an asymptotic treatment of a "stiff" phenomenon less accurate rather than more accurate. This follows because the disparity of time scales between fast and slow phenomena is often exploited in the asymptotic approach rather than tolerated. Furthermore, the non-convergence of any particular solution is often easier to spot in timestep splitting with asymptotics because the manner of degradation is usually catastrophic. In kinetics calculations, lack of conservation of mass or atoms signals inaccuracy rather clearly. 

Thus, there are secular terms in the asymptotic solution on the slow scale, which indicate that the approximation (4.4) is false for the long times T = 1/e. 

All three components x, y, z vary on the slow scale B = e t = e t. Singularities occur in the asymptotic solution at the finite time 6 = v. 

This slow crossover into asymptotic hydrodynamic behavior stands, interestingly enough (also for experiments), in marked contrast to the dynamic behavior of melts of short chains in this latter case, the corresponding length and time scales are sufficient to observe Rouse dynamics. For the case of solutions however, the chains have to be longer than in the melt case in order to observe the typical polymer behavior . Moreover, these results indicate that the observation of a decay of S k, t) alone is not sufficient to prove asymptotic Zimm behavior. 

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