A combination of two approaches to numerically solving second-order ODEs with a small parameter and singularities, such as interior and boundary layers, is considered, namely, compact high-order approximation schemes and explicit generation of layer resolving grids. The generation of layer resolving grids, which is based on estimates of solution derivatives and formulations of coordinate transformations eliminating the solution singularities, is a generalization of a method for a first-order scheme developed earlier. This paper presents formulas of the coordinate transformations and numerical experiments for first-, second-, and third-order schemes on uniform and layer resolving grids for equations with boundary, interior, exponential, and power layers of various scales. Numerical experiments confirm the uniform convergence of the numerical solutions performed with the compact high-order schemes on the layer resolving grids. By using transfinite interpolation or numerical solutions to the Beltrami and diffusion equations in a control metric based on coordinate transformations eliminating the solution singularities, this technology can be generalized to the solution of multi-dimensional equations with boundary and interior layers.

中文翻译：

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具有边界和内层问题的数值建模的紧凑差分方案和层解析网格

考虑使用两种方法来数值求解具有较小参数和奇异性的二阶ODE，例如内部层和边界层，即紧凑的高阶逼近方案和层生成网格的显式生成。基于解决方案导数的估计和消除解决方案奇异性的坐标转换公式的基础，层解析网格的生成是对先前开发的一阶方案的一种方法的概括。本文介绍了在边界层，内部层，指数层和幂阶层具有不同比例的方程组的均匀和层分辨网格上的一阶，二阶和三阶格式的坐标转换公式和数值实验。数值实验证实了在层解析网格上使用紧凑的高阶格式执行的数值解的一致收敛性。通过在基于坐标变换的控制度量中对Beltrami和扩散方程使用超限插值或数值解，消除了解决方案的奇异点，该技术可以推广到具有边界层和内部层的多维方程的解决方案。