Abstract

Layer-resolving grids remain important elements of software codes for solving real problems in layers with singularities, since they can substantially enhance the efficiency of computer resources. This paper describes an explicit approach to generating layer-resolving grids, which is aimed at using difference schemes of arbitrary orders. The approach is based on estimates of derivatives of solutions to singular perturbation problems; it is a generalization of an approach that has been developed for first-order schemes. The layer-resolving grids proposed are suitable to tackle problems with exponential-, power-, logarithmic-, and mixed-type boundary and interior layers. The theoretical results are confirmed by numerical experiments on a number of test problems with such layers performed with difference schemes of various orders of accuracy.

中文翻译：

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解决奇异摄动问题的自适应网格和高阶方案

摘要

层解析网格仍然是解决具有奇点的层中实际问题的软件代码的重要元素，因为它们可以显着提高计算机资源的效率。本文描述了一种生成层解析网格的显式方法，该方法旨在使用任意阶的差异方案。该方法基于对奇异扰动问题解的导数的估计；它是为一阶方案开发的方法的推广。提出的层解析网格适用于解决指数型、幂型、对数型和混合型边界层和内部层的问题。理论结果通过对许多测试问题的数值实验得到证实，这些层使用各种精度的不同方案执行。