Time scales asymptotic solutions

The solution of Eq. (69), and thus the asymptotic (f —> oo) behavior of the perturbation, is entirely specified by the eigenvectors and eigenvalues of the Jacobian matrix M. Assume all eigenvalues of M are ordered such that corresponds to the eigenvalue with maximal real part ) (/. i) > R(/-2) >. .. > The dominant time scales in response to the... 

The parameter , which enters into the downstream boundary conditions, can actually be interpreted as = T/,/71,., where T), - L2/D is the hydraulic diffusion time scale. When 1, there exists an intermediate asymptotic behavior corresponding to the zero-flux solution at z = L. When ( -C 1, the solution is essentially the membrane 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. 

It is attractive to seek solutions by use of activation-energy asymptotics. If there is a narrow reaction zone in the vicinity of x = 0, then by stretching the coordinate about x = 0 and excluding variations on a very short time scale, the augmented version of equation (56) becomes Xgd T/dx = — qgWi to the first approximation in the reaction zone. By use of equation (7-20), the integral of this equation across the reaction zone is seen to be expressible as... 

We have shown, using the method of matched asymptotic expansions, that in the outer domain there is an adjustable variable, and in the inner region there is another such variable. The composite solution is, therefore, a function of these two variables. Exploitation of this function is the essential idea behind the Multiple Time Scale method. Interested readers should refer to Nayfeh (1973) for exposition of this technique. 

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... 

This situation is standard for problems with a small parameter. The interval of fitness of the trivial asymptotic solution (4.1) cannot be too long. The small terms affect the solution for long times. On the long time scale t-lie one has to construct a new asymptotic solution depending on another typical time. A suitable new independent variable is t = et as one can guess by looking at the secular terms. 

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. 

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 ... 

The last remark concerns the general problem of identifying the typical magnitudes of dependent variables, x, y, z, especially in the case when some of the variables are zero at the initial moment. To make an error here is not fatal and it does not crash the asymptotics. The structure of the asymptotic solution compensates for any mistakes in the normalization, although a vagueness may occur when the time scales are defined. 

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... 

However, it is not hard to draw the phase portrait of the two-dimensional system (5.12). If we look at Fig. 2, we can see that there is a single stable equilibrium ((W/A), 0,0). Exactly the same one occurs in the original system (5.0). Therefore, the process described by (5.0) ends in this stable equilibrium and occurs on the time scale p. The exponents, which appear in the solution of the linearized equations, give the rate of asymptotic approximation to the stable state... 

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. 

If one considers either Problem (I) or (11), the question is to find some coefficients of the differential equations by comparing the solutions and the given (experimental) curves with each other. Since we use the explicit form of the asymptotic solutions, only those coefficients (or their combinations) can be found that are present in the formulas. In practice, the leading terms of the asymptotics provide an accuracy corresponding to the experimental accuracy. In Figures 3-5 both the experimental and theoretical curves for Problem (I) are presented on various time scales. 

Figure 3. Time scale II. CL method. Comparison between asymptotic solution-...

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