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

Abstract

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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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). https://doi.org/10.1134/S1995423913010023

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  • DOI: https://doi.org/10.1134/S1995423913010023

Keywords

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