Abstract:
This paper realizes a symbiosis of two approaches to the numerical solution of second order ODEs with
a small parameter having singularities such as interior and boundary layers, namely, the application of both
compact schemes of high orders and layer-resolving grids. The generation of layer-resolving grids, based on
estimates of solution derivatives and formulations of coordinate transformations eliminating solution singularities, is a generalization of the methodology early developed for the first order scheme.
This paper presents the formulas of the coordinate transformations and numerical experiments for the
schemes of the first, second, and third orders on uniform and layer-resolving grids for the equations with
boundary, interior, exponential and power layers of the first and second scales. The experiments conducted
confirm the uniform convergence of the numerical solutions of equations with the help of compact schemes of
high orders on the layer-resolving grids.
By using the transfinite interpolation methodology or numerical solutions to the Beltrami and diffusion
equations in a control metric, built by the coordinate transformations eliminating the solution singularities,
the developed technology can be generalized to the solution of multi-dimensional equations with boundary
and interior layers.
Key words:
equation with a small parameter, boundary layer, interior layer, compact scheme, scheme of high order, layer-resolving grid, adaptive grid.
Citation:
V. D. Liseikin, V. I. Paasonen, “Compact difference schemes and layer-resolving grids for the numerical modeling of problems with boundary and interior layers”, Sib. Zh. Vychisl. Mat., 22:1 (2019), 41–56; Num. Anal. Appl., 12:1 (2019), 37–50
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\by V.~D.~Liseikin, V.~I.~Paasonen
\paper Compact difference schemes and layer-resolving grids for the numerical modeling of problems with boundary and interior layers
\jour Sib. Zh. Vychisl. Mat.
\yr 2019
\vol 22
\issue 1
\pages 41--56
\mathnet{http://mi.mathnet.ru/sjvm700}
\crossref{https://doi.org/10.15372/SJNM20190104}
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\transl
\jour Num. Anal. Appl.
\yr 2019
\vol 12
\issue 1
\pages 37--50
\crossref{https://doi.org/10.1134/S199542391901004X}
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Linking options:
https://www.mathnet.ru/eng/sjvm700
https://www.mathnet.ru/eng/sjvm/v22/i1/p41
This publication is cited in the following 4 articles:
Navnit Jha, Shikha Verma, “A High-Resolution Convergent Radial Basis Functions Compact-FDD for Boundary Layer Problems on a Scattered Mesh Network Appearing in Viscous Elastic Fluid”, Int. J. Appl. Comput. Math, 8:5 (2022)
A. N. Kudryavtsev, V. D. Liseikin, A. V. Mukhortov, “An analysis of grid-clustering rules in a boundary layer using the numerical solution of the problem of viscous flow over a plate”, Comput. Math. Math. Phys., 62:8 (2022), 1356–1371
V. D. Liseikin, V. I. Paasonen, “Adaptive grids and high-order schemes for solving singularly-perturbed problems”, Num. Anal. Appl., 14:1 (2021), 69–82
V. D. Liseikin, S. Karasuljic, A. V. Mukhortov, V. I. Paasonen, Lecture Notes in Computational Science and Engineering, 143, Numerical Geometry, Grid Generation and Scientific Computing, 2021, 227
Citation in format AMSBIB
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