Robust difference scheme for the Cauchy problem for a singularly perturbed ordinary differential equation

    Lidia Pavlovna Shishkina Affiliation
    ; Grigorii Ivanovich Shishkin Affiliation


Grid approximation of the Cauchy problem on the interval  D = {0 ≤ x ≤ d} is first studied for a linear singularly perturbed ordinary differential equation of the first order with a perturbation parameter ε multiplying the derivative in the equation where the parameter ε takes arbitrary values in the half-open interval (0, 1]. In the Cauchy problem under consideration, for small values of the parameter ε, a boundary layer of width O(ε) appears on which the solution varies by a finite value. It is shown that, for such a Cauchy problem, the solution of the standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm; convergence occurs only under the condition h ε, where h = d N −1 , N is the number of grid intervals, h is the grid step-size. Taking into account the behavior of the singular component in the solution, a special piecewise-uniform grid is constructed that condenses in a neighborhood of the boundary layer. It is established that the standard difference scheme on such a special grid converges ε-uniformly in the maximum norm at the rate O(N −1 lnN). Such a scheme is called a robust one.

For a model Cauchy problem for a singularly perturbed ordinary differential equation, standard difference schemes on a uniform grid (a classical difference scheme) and on a piecewise-uniform grid (a special difference scheme) are constructed and investigated. The results of numerical experiments are given, which are consistent with theoretical results.

Keyword : singularly perturbed Cauchy problem, ordinary differential equation, boundary layer, a priori estimates, standard difference scheme, uniform grid, piecewise-uniform grid, maximum norm, solution decomposition, robust difference scheme

How to Cite
Shishkina, L., & Shishkin, G. (2018). Robust difference scheme for the Cauchy problem for a singularly perturbed ordinary differential equation. Mathematical Modelling and Analysis, 23(4), 527-537.
Published in Issue
Oct 9, 2018
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[1] P.W. Hemker, G.I. Shishkin and L.P. Shishkina. The use of defect correction for the solution of parabolic singular perturbation problems. ZAMM – Z. Angew. Math. Mech. , 77 (1):59–74, 1997.

[2] P.W. Hemker, G.I. Shishkin and L.P. Shishkina. ε-uniform schemes with high-order time-accuracy for parabolic singular perturbation problems. IMA J. Numer. Anal.,
20 (1):99–121, 2000.

[3] N.N. Kalitkin. Numerical methods. Nauka, Moscow, 1978. (in Russian)

[4] N.N. Kalitkin and P.V. Koriakin. Numerical methods. Methods of mathematical physics. Akademia, Moscow, 2013. (in Russian)

[5] G.I. Marchuk and V.V. Shaidurov. Difference Methods and Their Interpolations. Springer–Verlag, New York, 1983.

[6] J. J. H. Miller, E. O’Riordan and G.I. Shishkin. Fitted numerical methods for singular perturbation problems. Error estimates in the maximum norm for linear problems in one and two dimensions, Revised Edition. World Scientific, Singapore, 2012.

[7] A. Quarteroni, R. Sacco and F. Saleri. Numerical Mathematics, Second Edition. Springer, Berlin Heidelberg, 2007.

[8] H. G. Roos, M. Stynes and L. Tobiska. Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion-Reaction and Flow Problems. Second Edition. Marcel Dekker, Springer-Verlag, Berlin, 2008.

[9] A. A. Samarskii. The Theory of Difference Schemes. Marcel Dekker, New York, 2001.

[10] G. I. Shishkin. Discrete Approximations of Singularly Perturbed Elliptic and Parabolic Equations. Russian Academy of Sciences, Ural Section, Ekaterinburg, 1992. (in Russian)

[11] G.I. Shishkin and L.P Shishkina. Difference Methods for Singular Perturbation Problems. Monographs & Surveys in Pure & Applied Mathematics. Chapman and Hall/CRC, Boca Raton, 2009.

[12] E. Suli and D. Mayers. An Introduction to Numerical Analysis. Cambridge University Press, Cambridge, 2003. Math. Model. Anal., 23(4):527–537, 2018.