Topics in Singular Perturbations and Hybrid Asymptotic-Numerical Methods

Hybrid asymptotic-numerical methods are used to study various singular perturbation problems where innnite logarithmic asymptotic expansions occur. In particular, they are used to sum the innnite logarithmic expansions that arise for a low Reynolds number ow problem and for certain two-dimensional eigenvalue problems in perforated domains. Asymptotic and numerical methods are also used to analyze the exponentially slow internal layer motion that occurs for certain phase separation models and for other nonlinear diiusive problems with exponentially ill-conditioned linearizations. The study of this dynamical metastability phenomena involves exponential asymptotics. 0 Introduction The method of matched asymptotic expansions is a powerful systematic analytical method for asymptotically calculating solutions to singularly perturbed problems. It has been successfully used in a wide variety of applications (see H], KC], LA], OR], V]). However, there are certain special classes of problems where this method has some apparent limitations. In particular, for problems involving innnite logarithmic series in powers of = ?1= log ", where " is a small positive parameter, it is well-known that this method may be of only limited practical use in approximating the exact solution accurately. This diiculty stems from the fact that ! 0 very slowly as " decreases. Therefore, unless many coeecients in the innnite logarithmic series can be obtained analytically, the resulting low order truncation of this series will typically not be very accurate unless " is very small. Singular perturbation problems involving innnite logarithmic expansions arise in many areas of application including; low Reynolds number uid ow past cylindrical bodies and related model problems HS] thermal runaway behavior for cylindrical chemical reactors containing cooling rods LW91], WK91], WHK], LW94]; two-dimensional diiusion problems in singularly perturbed regions TG], TW]. In x1-2 we will summarize and extend some recent work of KWK], KW], TW], and WHK] where hybrid asymptotic-numerical methods are used to sum innnite logarithmic expansions in various contexts. Examples of such problems that will be considered include the calculation of the drag coeecient for slow viscous ow past a cylinder and the determination of eigenvalue parameters for some linear and nonlinear eigenvalue problems in perforated domains. Another class of problems where diiculties with the method of matched asymp-totic expansions arise is for problems where exponentially small terms of the form O e ?c=" , for " ! 0 with c > 0, need to be captured analytically in order to characterize the phenomena under consideration. For such problems in …