On exponential ill-conditioning and internal layer behavior

In the limit of small diffusivity, the internal layer behavior associated with the initial-boundary value problems for a viscous shock equation and a reaction diffusion equation is analyzed. As a result of the occurrence of exponentially small eigenvalues for the linearized problems, the steady state internal layer solutions are shown to very sensitive to small perturbations. For the time dependent problems, the small eigenvalues give rise to exponentially slow internal layer motion. Accurate numerical methods are used to compute the steady state internal layer solutions and the slow internal layer motion. The relationship between the viscous shock problem and some exponentially ill-conditioned linear singular perturbation problems is discussed. © 1995, Taylor & Francis Group, LLC. All rights reserved.