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Solving a second-order nonlinear singular perturbation ordinary differential equation by a Samarskii scheme


A boundary value problem for a second-order nonlinear singular perturbation ordinary differential equation is considered. A method based on Newton and Picard linearizations using a modified Samarskii scheme on a Shishkin grid for a linear problem is proposed. It is proved that the difference schemes are of second-order and uniformly convergent. To decrease the number of arithmetic operations, a two-grid method is proposed. The results of some numerical experiments are discussed.

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  1. Ilyin, A.M., A Difference Scheme for a Differential Equation with a Small Parameter at the Highest Derivative, Mat. Zametki, 1969, vol. 6, no. 2, pp. 237–248.

    MathSciNet  Google Scholar 

  2. Bakhvalov, N.S., On Optimization of Methods to Solve Boundary Value Problems in the Presence of a Boundary Layer, Zh. Vych. Mat. Mat. Fiz., 1969, vol. 9, no. 4, pp. 841–859.

    MATH  Google Scholar 

  3. Shishkin, G.I., Setochnye approksimatsii singulyarno vozmushchyonnykh ellipticheskikh i parabolicheskikh uravnenii (Grid Approximations of Singular Perturbation Elliptic and Parabolic Equations), Yekaterinburg: UB RAS, 1992.

    Google Scholar 

  4. Farrell, P.A., Hegarty, A.F., Miller, J.J., O’Riordan, E., and Shishkin, G.I., Robust Computational Techniques for Boundary Layers, Boca Raton, FL: Chapman and Hall, CRC Press, 2000.

    MATH  Google Scholar 

  5. Bagayev, B.M., Karepova, E.D., and Shaidurov, V.V., Setochnye metody resheniya zadach s pogranichnym sloem, ch. 2 (Grid Methods to Solve Problems with a Boundary Layer. Part. 2), Novosibirsk: Nauka, 2001.

    Google Scholar 

  6. Shishkin, G.I., A High-Accuracy Method for a Quasi-Linear Singularly Perturbed Elliptic Convection-Diffusion Equation, Zh. Vych. Mat., 2006, vol. 9, no. 1, pp. 81–108.

    MATH  Google Scholar 

  7. Roos, H.-G., Stynes, M., and Tobiska, L., Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems, Springer Ser. Comp. Math., vol. 24, Berlin: Springer-Verlag, 1996.

    Google Scholar 

  8. Andreev, V.B. and Savin, I.A., On the Convergence, Uniform with Respect to a Small Parameter, of A.A. Samarskii’s Monotone Scheme and Its Modifications, Zh. Vych. Mat. Mat. Fiz., 1995, vol. 35, no. 5, pp. 739–752.

    MathSciNet  Google Scholar 

  9. Vulanovic, R., A Uniform Numerical Method for Quasilinear Singular Perturbation Problems without Turning Points, Computing, 1989, vol. 41, pp. 97–106.

    MathSciNet  MATH  Article  Google Scholar 

  10. Vulkov, L.G. and Zadorin, A.I., Two-Grid Algorithms for an Ordinary Second Order Equation with Exponential Boundary Layer in the Solution, Int. J. Num. Anal. Model., 2010, vol. 7, no. 3, pp. 580–592.

    MathSciNet  MATH  Google Scholar 

  11. Zadorin, A.I., An Interpolation Method on a Condensing Grid for a Function with a Boundary Layer Component, Zh. Vych. Mat. Mat. Fiz., 2008, vol. 48, no. 9, pp. 1673–1684.

    MathSciNet  Google Scholar 

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Correspondence to A. I. Zadorin.

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Original Russian Text © A.I. Zadorin, S.V. Tikhovskaya, 2013, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2013, Vol. 16, No. 1, pp. 11–25.

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Zadorin, A.I., Tikhovskaya, S.V. Solving a second-order nonlinear singular perturbation ordinary differential equation by a Samarskii scheme. Numer. Analys. Appl. 6, 9–23 (2013).

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  • second-order nonlinear ordinary differential equation
  • singular perturbation
  • Newton method
  • Picard method
  • Samarskii scheme
  • Shishkin grid
  • uniform convergence
  • two-grid algorithm