Basic Concepts Underlying Singular Perturbation Techniques

Abstract

Summary. In many singular perturbation problems multiple scales are used. For instance, one may use both the coordinate x and the coordinate x* e-ix. In a secular-type problem x and x* are used simultaneously. This paper discusses layer-type problems in which x* is used in a thin layer and x outside this layer. Assume one seeks approximations to a function f(x, e), uniformly valid to some order in for x in a closed interval D. In layer-type problems one uses (at least) two expansions (called inner and outer) neither of which is uniformly valid but whose domains of validity together cover the interval D. To define "domain of validity" one needs to consider intervals whose endpoints depend on e. In the construction of the inner and outer expansions, constants and functions of occur which are determined by comparison of the two expansions "matching." The comparison is possible only in the domain of overlap of their regions of validity. Once overlap is established, matching is easily carried out. Heuristic ideas for determining domains of validity of approximations by a study of the corresponding equations are illustrated with the aid of model equations. It is shown that formally small terms in an equation may have large integrated effects. The study of this is of central importance for understanding layer-type problems. It is emphasized that considering the expansions as the result of applying limit processes can lead to serious errors and, in any case, hides the nature of the expansions.

The height of maximum speed-up in the atmospheric boundary layer flow over low hills

“…This analysis applied to Jackson and Hunt's (1975) solution, for example, does not produce a simple expression for l because their speed-up profile depends on a function that is unknown in the general case. On the other hand, the results proposed by Taylor and Lee (1984), Lemelin et al (1988) and Finnigan (1992) yield a rather straightforward calculation. In a recent study, Pellegrini and Bodstein (2002a) also obtained an expression for the height of maximum speed-up based on the analytical speed-up function they proposed on a companion paper (Pellegrini and Bodstein, 2002b).…”

“…Further analysis of the equation for l indicates that the predicted values can also be used to estimate l max . We also calculate expressions for l max from the analytical speed-up profiles proposed by Taylor and Lee (1984), Lemelin et al (1988), Finnigan (1992) and Pellegrini and Bodstein (2002b). The resulting expressions for l max are compared to field data, and it can be seen that the results obtained from the profiles of Taylor and Lee (1984) and Lemelin et al (1988) are rather similar to that of JH.…”

One-dimensional solidification of supercooled melts

“…Let us now consider the governing equations (14)- (17). Noting that F (y, 0) = −1 we look for a small time similarity solution close to F = −1 and so re-define the function…”

“…The factor t 1/2 in front of G(η) in (28) is chosen to balance the Stefan condition in (30) and then it follows that s ∝ t. This leaves time in the governing equation (29) and the boundary condition (30a): for this reason we make the small time substitution t = τ , s = r, where is an artificial small parameter [16,17], to find…”

Matched asymptotic expansion calculation of the equilibrium shape of a hole in a thin liquid film

“…This is a well-known phenomenon indicating that there are terms missing called "switchbacks" (see [3]). Essentially the form of the upper series is incorrect and instead of looking for a solution in the form C = Co + £2Ci> we should seek £ = lneCswi + Co + £2Ci-Substitution of this revised series into the inner equation yields the result dC"…”