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A higher order accurate solution decomposition scheme for a singularly perturbed parabolic reaction-diffusion equation

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Abstract

An initial-boundary value problem is considered for a singularly perturbed parabolic reaction-diffusion equation. For this problem, a technique is developed for constructing higher order accurate difference schemes that converge ɛ-uniformly in the maximum norm (where ɛ is the perturbation parameter multiplying the highest order derivative, ɛ ∈ (0, 1]). A solution decomposition scheme is described in which the grid subproblems for the regular and singular solution components are considered on uniform meshes. The Richardson technique is used to construct a higher order accurate solution decomposition scheme whose solution converges ɛ-uniformly in the maximum norm at a rate of , where N + 1 and N 0 + 1 are the numbers of nodes in uniform meshes in x and t, respectively. Also, a new numerical-analytical Richardson scheme for the solution decomposition method is developed. Relying on the approach proposed, improved difference schemes can be constructed by applying the solution decomposition method and the Richardson extrapolation method when the number of embedded grids is more than two. These schemes converge ɛ-uniformly with an order close to the sixth in x and equal to the third in t.

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References

  1. G. I. Marchuk and V. V. Shaidurov, Improvement of the Accuracy of Solutions to Difference Schemes (Nauka, Moscow, 1979) [in Russian].

    Google Scholar 

  2. P. W. Hemker, G. I. Shishkin, and L. P. Shishkina, “High-order accurate decomposition of Richardson’s method for a singularly perturbed elliptic reaction-diffusion equation,” Comput. Math. Math. Phys. 44(2), 309–316 (2004).

    MathSciNet  Google Scholar 

  3. L. P. Shishkina, “The Richardson method of high-order accuracy in t for a semilinear singularly perturbed parabolic reaction-diffusion equation on a strip,” Proceedings of the International Conference on Computational Mathematics ICCM’2004, Novosibirsk, June 2004 (ICM & MG, Novosibirsk, 2004), Part II, pp. 927–931.

    Google Scholar 

  4. G. I. Shishkin and L. P. Shishkina, “A higher-order Richardson method for a quasilinear singularly perturbed elliptic reaction-diffusion equation,” Differ. Equations 41(7), 1030–1039 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  5. G. I. Shishkin, “Robust novel high-order accurate numerical methods for singularly perturbed convection-diffusion problems,” Math. Model. Anal. 10(4), 393–412 (2005).

    MATH  MathSciNet  Google Scholar 

  6. G. I. Shishkin, “Richardson’s method for increasing the accuracy of difference solutions of singularly perturbed elliptic convection-diffusion equations,” Russ. Math. 50(2), 57–71 (2006).

    MathSciNet  Google Scholar 

  7. G. I. Shishkin, “The Richardson scheme for the singularly perturbed parabolic reaction-diffusion equation in the case of a discontinuous initial condition,” Comput. Math. Math. Phys. 49(8), 1348–1368 (2009).

    Article  MathSciNet  Google Scholar 

  8. G. I. Shishkin and L. P. Shishkina, “A higher order Richardson scheme for a singularly perturbed semilinear elliptic convection-diffusion equation,” Comput. Math. Math. Phys. 50(3), 437–456 (2010).

    Article  MathSciNet  Google Scholar 

  9. G. I. Shishkin and L. P. Shishkina, Difference Methods for Singular Perturbation Problems (Chapman and Hall/CRC, Boca Raton, 2009).

    MATH  Google Scholar 

  10. G. I. Shishkin and L. P. Shishkina, “Improved finite difference scheme of the solution decomposition method for a singularly perturbed reaction-diffusion equation,” Proc. Steklov Inst. Math. 272(1), Suppl. 197–214 (2010).

    Google Scholar 

  11. A. M. Il’in, “Differencing scheme for a differential equation with a small parameter affecting the highest derivative,” Math. Notes 6, 596–602 (1969).

    Article  Google Scholar 

  12. D. N. Allen and R. V. Southwell, “Relaxation methods applied to determine the motion, in two dimensions, of viscous fluid past a fixed cylinder,” Quart. J. Mech. Appl. Math. 8(2), 129–145 (1955).

    Article  MATH  MathSciNet  Google Scholar 

  13. N. S. Bakhvalov, “The optimization of methods of solving boundary value problems with a boundary layer,” USSR Comput. Math. Math. Phys. 9(4), 139–166 (1969).

    Article  Google Scholar 

  14. J. J. H. Miller and E. O’Riordan, “Necessity of fitted operators and Shishkin meshes for resolving thin layer phenomena,” CWI Quarterly 10(3/4), 207–213 (1997).

    MATH  MathSciNet  Google Scholar 

  15. G. I. Shishkin, “Approximation of solutions of singularly perturbed boundary value problems with a parabolic boundary layer,” USSR Comput. Math. Math. Phys. 29(4), 1–10 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  16. G. I. Shishkin, “Difference scheme of the solution decomposition method for a singularly perturbed parabolic reaction-diffusion equation,” Russ. J. Numer. Anal. Math. Model. 25(3), 261–278 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  17. G. I. Shishkin and L. P. Shishkina, “A Richardson scheme of the decomposition method for solving singularly perturbed parabolic reaction-diffusion equation,” Comput. Math. Math. Phys. 50, 2003–2022 (2010).

    Article  MathSciNet  Google Scholar 

  18. H. Schlichting, Boundary Layer Theory (McGraw-Hill, New York, 1979).

    MATH  Google Scholar 

  19. G. I. Shishkin and L. P. Shishkina, “Improved approximations of the solution and derivatives to a singularly perturbed reaction-diffusion equation based on the solution decomposition method,” Comput. Math. Math. Phys. 51(6), 1020–1049 (2011).

    Article  MathSciNet  Google Scholar 

  20. A. A. Samarskii, The Theory of Finite Difference Schemes (Nauka, Moscow, 1989; Marcel Dekker, New York, 2001).

    Google Scholar 

  21. A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First-Order Partial Differential Equations (Fizmatlit, Moscow, 2001; Taylor and Francis, London, 2002).

    Book  Google Scholar 

  22. N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel’kov, Numerical Methods (Laboratoriya Bazovykh Znanii, Moscow, 2001) [in Russian].

    Google Scholar 

  23. G. I. Shishkin, Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations (Ural Otd. Ross. Akad. Nauk, Yekaterinburg, 1992) [in Russian].

    Google Scholar 

  24. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type (Nauka, Moscow, 1967; Am. Math. Soc., Providence, R.I., 1968).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620990, Russia

    G. I. Shishkin & L. P. Shishkina

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  1. G. I. Shishkin
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  2. L. P. Shishkina
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Correspondence to G. I. Shishkin.

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Original Russian Text © G.I. Shishkin, L.P. Shishkina, 2015, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2015, Vol. 55, No. 3, pp. 393–416.

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Shishkin, G.I., Shishkina, L.P. A higher order accurate solution decomposition scheme for a singularly perturbed parabolic reaction-diffusion equation. Comput. Math. and Math. Phys. 55, 386–409 (2015). https://doi.org/10.1134/S0965542515030161

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