Adaptive grids and high-order schemes for solving singularly-perturbed problems

Layer-resolving grids remain an important element of comprehensive software codes when solving real-life problems with layers of singularities as they can substantially enhance the efficiency of computer-resource utilization. This paper describes an explicit approach to generating layer-resolving grids which is aimed at application of difference schemes of an arbitrary order. The approach proposed is based on estimates of derivatives of solutions to singularly-perturbed problems and is a generalization of the approach developed for the first order schemes. The layer-resolving grids proposed are suitable to tackle problems with exponential-, power-, logarithmic-, and mixed-type boundary and interior layers. Theoretical results have been confirmed by the numerical experiments on a number of test problems with such layers; the results were compared to those obtained with difference schemes of different orders of accuracy.