Hyperasymptotics and the Linear Boundary Layer Problem: Why Asymptotic Series Diverge

Abstract

Abstract. The simplest problem with boundary layers, 2 uxx − u = −f (x), is used to illustrate (i) why the perturbation series in powers of is asymptotic but divergent, (ii) why the optimally truncated expansion is "superasymptotic" in the sense that that error is proportional to exp(−[constant]/ ), and (iii) how to obtain an improved "hyperasymptotic" approximation.

Three-dimensional capillary waves due to a submerged source with small surface tension

“…The variables β and τ determine the relationship between the Froude and Weber number. This gives a system that is nearly identical to (8)- (12), with the dynamic condition (10) now given by β 2 (|∇φ 2 | − 1) + ξ = βτ 2 κ on z = ξ(x, y).…”

“…These techniques, described in [7], [6], [41], [14], [5], and elsewhere, have been developed for the purpose of calculating such exponentially small behaviour. More general introductions to exponential asymptotic methods are found in [42], [10] and [33], while comprehensive discussions are found in [45] and [12]. In section 1.3, we will describe the exponential asymptotic methodology used in the present study, which is based on the work of [41], [14], [15], and the theory in [31].…”

Nanoptera in a Period-2 Toda Chain

“…This makes the technique particularly useful for the many nonlinear problems for which obtaining even these low-order correction terms is intractable. See the review article [14] or monograph [13] for more details on exponential asymptotics and their application to nonlocal solitary waves, [8,9,16] for examples of previous studies of exponential asymptotics, and [20,61] for more details on the particular methodology that we apply in the present paper.…”

RBF-vortex methods for the barotropic vorticity equation on a sphere

“…A WKB-argument provides a more precise answer: when a wave of wavescale L propagates through a slowly-varying medium whose scale of variation is M -whether the variations are due to changes in numerics or the physical index of refraction -then the reflection is proportional to exp(−q/ ) where = L/M and the constant q can be calculated only by some form of hyperasymptotic ("beyond-all-orders") perturbation theory [10,11,13]. Spurious reflection can thus be controlled -made exponentially small -by varying the grid sufficiently slowly.…”