Difference Scheme on a Uniform Grid for the Singularly Perturbed Cauchy Problem
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We consider the Cauchy problem for a singularly perturbed second order ordinary differential equation. Based on the maximum principle, we estimate the solution and its derivatives. We construct an exponential fitted scheme generalizing the well-known scheme due to A. M. Il’in. We also prove the uniform convergence with the first order accuracy and illustrate the results by numerical experiments. Bibliography: 8 titles.
KeywordsBoundary Layer Cauchy Problem Maximum Principle Uniform Convergence Uniform Grid
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- 1.N. S. Bakhvalov, “The optimization of methods of solving boundary value problems with a boundary layer” [in Russian], Zh. Vychisl. Mat. Mat. Fiz. 9, No. 4, 841–859 (1969); English transl.: U.S.S.R. Comput. Math. Math. Phys. 9, No. 4, 139–166 (1971).Google Scholar
- 2.G. I. Shishkin, Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations [in Russian], Ekaterinoburg (1992).Google Scholar
- 3.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, World Scientific, Singapore (1996).Google Scholar
- 4.A. M. Il’in, “Difference scheme for a differential equation with a small parameter affecting the highest derivative” [in Russian], Mat. Zametki 6, No. 2, 237–248 (1969); English transl.: Math. Notes 6, No. 2, 596–602 (1969).Google Scholar
- 5.E. P. Doolan, J. J. H. Miller, and W. H. A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers Boole Press, Dublin (1980).Google Scholar
- 6.A. A. Samarskii, The Theory of Difference Schemes [in Russian], Nauka, Moscow (1983); English transl.: Marcel Dekker, New York (2001).Google Scholar
- 8.G. I. Marchuk and V. V. Shaidurov, Increasing the Accuracy of Solutions of Difference Schemes [in Russian], Nauka, Moscow (1979); English transl.: Difference Methods and Their Extrapolations, Springer, New York etc. (1983).Google Scholar