Fast time scales, asymptotic solution

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. 

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... 

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