TY - JOUR

T1 - Compact Difference Schemes and Layer Resolving Grids for Numerical Modeling of Problems with Boundary and Interior Layers

AU - Liseikin, V. D.

AU - Paasonen, V. I.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - adaptive grid

KW - boundary layer

KW - compact scheme

KW - equation with a small parameter

KW - high-order scheme

KW - interior layer

KW - layer resolving grid

UR - http://www.scopus.com/inward/record.url?scp=85063995069&partnerID=8YFLogxK

U2 - 10.1134/S199542391901004X

DO - 10.1134/S199542391901004X

M3 - Article

AN - SCOPUS:85063995069

VL - 12

SP - 37

EP - 50

JO - Numerical Analysis and Applications

JF - Numerical Analysis and Applications

SN - 1995-4239

IS - 1

ER -