Numerical Analysis and Applications

, Volume 12, Issue 1, pp 37–50 | Cite as

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

  • V. D. LiseikinEmail author
  • V. I. Paasonen


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.


equation with a small parameter boundary layer interior layer compact scheme high-order scheme layer resolving grid adaptive grid 


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  1. 1.
    Liseikin, V.D., Estimates for Derivatives of Solutions to Differential Equations with Boundary and Interior Layers, Sib.Mat. Zh., 1992, vol. 33, no. 6, pp. 106–117.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Liseikin, V.D., Layer Resolving Grids and Transformations for Singular Perturbation Problems, Utrecht: VSP, 2001.CrossRefGoogle Scholar
  3. 3.
    Liseikin, V.D., Grid Generation Methods, 3rd ed., Berlin: Springer, 2017.CrossRefzbMATHGoogle Scholar
  4. 4.
    Paasonen, V.I., Compact Third-Order Accuracy Schemes on Non-Uniform Adaptive Grids, Vych. Tekhnol., 2015, vol. 20, no. 2, pp. 56–64.zbMATHGoogle Scholar
  5. 5.
    Paasonen, V.I., A Third-Order Approximation Scheme on a Non-Uniform Grid for Navier–Stokes Equations, Vych. Tekhnol., 2000, vol. 5, no. 2, pp. 78–85.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Glukhovskii, A.S. and Paasonen, V.I., Compact Difference Schemes for Navier–Stokes Equations on Non-UniformGrids,Marchukovskie nauchnye chteniya 2017 (Marchuk ScientificReadings 2017), Novosibirsk: Publ. House of ICM&MG SB RAS, 2017, pp. 211–217.Google Scholar
  7. 7.
    Liseikin, V.D., Numerical Solution of Equations with Power Boundary Layer, Zh. Vych. Mat. Mat. Fiz., 1986, vol. 26, no. 12, pp. 1813–1820.MathSciNetGoogle Scholar
  8. 8.
    Bakhvalov, N.S., On the Optimization of the Methods for Solving Boundary Value Problems in the Presence of a Boundary Layer, Zh. Vych. Mat. Mat. Fiz., 1969, vol. 9, no. 4, pp. 842–859.MathSciNetGoogle Scholar
  9. 9.
    Shishkin, G.I., A Difference Scheme for a Singularly Perturbed Equation of Parabolic Type with a Discontinuous Initial Condition, Dokl. Akad. Nauk SSSR, 1988, vol. 37, pp. 792–796.MathSciNetzbMATHGoogle Scholar
  10. 10.
    Liseikin, V.D., Numerical Solution of a Singularly Perturbed Equation with a Turning Point, Zh. Vych. Mat. Mat. Fiz., 1984, vol. 24, no. 12, pp. 1812–1818.MathSciNetGoogle Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Institute of Computational Technologies, Siberian BranchRussian Academy of SciencesNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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