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Advances in Applied Mathematics pp 139–151Cite as

A Technique to Construct Grid Methods of Higher Accuracy Order for a Singularly Perturbed Parabolic Reaction-Diffusion Equation

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 87))

Abstract

We consider a technique to construct \(\varepsilon\)-uniformly convergent in the maximum norm grid approximations of higher accuracy order on uniform grids for a singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter \(\varepsilon\) (\(\varepsilon \in (0,1]\)) multiplying the highest-order derivative, the solution of which has a parabolic boundary layer in a neighborhood of the lateral boundary.

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Notes

  1. 1.

    The notation \(L_{(j.k)}\ (M_{(j.k)},\ G_{h(j.k)})\) means that these operators (constants, grids) are introduced in formula (j. k).

  2. 2.

    By M (or m), we denote sufficiently large (small) positive constants independent of the parameter \(\varepsilon\) and of the discretization parameters.

References

  1. Marchuk, G.I., Shaidurov, V.V.: Difference Methods and Their Interpolations. Springer, New York (1983)

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  2. Shishkin, G.I., Shishkina, L.P.: Difference Methods for Singular Perturbation Problems. Monographs and Surveys in Pure and Applied Mathematics. Chapman and Hall/CRC, Boca Raton (2009)

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  3. Shishkin, G.I., Shishkina, L.P.: A Richardson scheme of the decomposition method for solving singularly perturbed parabolic reaction-diffusion equation. Comp. Math. Math. Phys. 50(12), 2003–2022 (2010)

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  4. Bakhvalov, N.S.: Numerical Methods. Nauka, Moscow (1973) (in Russian)

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  5. Marchuk, G.I.: Methods of Numerical Mathematics. Nauka, Moscow (1989) (in Russian)

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Acknowledgements

This research was supported by the Russian Foundation for Basic Research under grant no.13-01-00618.

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Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Russian Academy of Sciences, Ekaterinburg, Russia

    L. Shishkina & G. Shishkin

Authors
  1. L. Shishkina
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  2. G. Shishkin
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Correspondence to L. Shishkina .

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Ali R. Ansari

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Shishkina, L., Shishkin, G. (2014). A Technique to Construct Grid Methods of Higher Accuracy Order for a Singularly Perturbed Parabolic Reaction-Diffusion Equation. In: Ansari, A. (eds) Advances in Applied Mathematics. Springer Proceedings in Mathematics & Statistics, vol 87. Springer, Cham. https://doi.org/10.1007/978-3-319-06923-4_13

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