An analysis of a uniformly accurate difference method for a singular perturbation problem
Authors:
Alan E. Berger, Jay M. Solomon and Melvyn Ciment
Journal:
Math. Comp. 37 (1981), 7994
MSC:
Primary 65L10
MathSciNet review:
616361
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Abstract: It will be proven that an exponential tridiagonal difference scheme, when applied with a uniform mesh of size h to: for , b and f smooth, in (0, 1], and and given, is uniformly secondorder accurate (i.e., the maximum of the errors at the grid points is bounded by with the constant C independent of h and ). This scheme was derived by ElMistikawy and Werle by a patching of a pair of piecewise constant coefficient approximate differential equations across a common grid point. The behavior of the approximate solution in between the grid points will be analyzed, and some numerical results will also be given.
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 A. E. Berger, J. M. Solomon & M. Ciment, "Higher order accurate tridiagonal difference schemes for diffusion convection equations," Advances in Computer Methods for Partial Differential EquationsIII (R. Vichnevetsky and R. S. Stepleman, Eds.), Proc. Third IMACS Conference on Computer Methods for Partial Differential Equations, June 1979, Lehigh University, pp. 322330. MR 603482 (82b:65149)
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 A. E. Berger, J. M. Solomon & M. Ciment, Uniformly Accurate Difference Methods for a Singular Perturbation Problem, Proc. Internat. Conf. on Boundary and Interior Layers, Computational and Asymptotic Methods, June 36, 1980, Trinity College, Dublin, Ireland (J. J. H. Miller, Ed.), Boole Press, Dublin, 1980, pp. 1428. MR 589348 (82e:65083)
 [3]
 A. E. Berger, J. M. Solomon, M. Ciment, S. H. Leventhal & B. C. Weinberg, "Generalized operator compact implicit schemes for boundary layer problems," Math. Comp., v. 35, 1980, pp. 695731. MR 572850 (81f:65057)
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 T. M. ElMistikawy & M. J. Werle, "Numerical method for boundary layers with blowingThe exponential box scheme," AIAA J., v. 16, 1978, pp. 749751.
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 P. P. N. de Groen & P. W. Hemker, "Error bounds for exponentially fitted Galerkin methods applied to stiff twopoint boundary value problems," Numerical Analysis of Singular Perturbation Problems (P. W. Hemker and J. J. H. Miller, Eds.), Academic Press, New York, 1979, pp. 217249. MR 556520 (81a:65076)
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 A. F. Hegarty, J. J. H. Miller & E. O'Riordan, Uniform Second Order Difference Schemes for Singular Perturbation Problems, Proc. Internat. Conf. on Boundary and Interior Layers, Computational and Asymptotic Methods, June 36, 1980, Trinity College, Dublin, Ireland (J. J. H. Miller, Ed.), Boole Press, Dublin, 1980, pp. 301305. MR 589380 (83h:65095)
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 A. M. Il'in, "Differencing scheme for a differential equation with a small parameter affecting the highest derivative," Mat. Zametki, v. 6, 1969, pp. 237248 = Math. Notes, v. 6, 1969, pp. 596602. MR 0260195 (41:4823)
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 R. B. Kellogg & A. Tsan, "Analysis of some difference approximations for a singular perturbation problem without turning points," Math. Comp., v. 32, 1978, pp. 10251039. MR 0483484 (58:3485)
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 J. Lorenz, Stability and Consistency Analysis of Difference Methods for Singular Perturbation Problems, Proc. Conf. on Analytical and Numerical Approaches to Asymptotic Problems in Analysis, June 913, 1980, University of Nijmegen, The Netherlands (O. Axelsson, L. Frank and A. Van der Sluis, Eds.), NorthHolland, Amsterdam, 1981. MR 605505 (83b:65077)
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 J. J. H. Miller, "Sufficient conditions for the convergence, uniformly in epsilon, of a three point difference scheme for a singular perturbation problem," Numerical Treatment of Differential Equations in Applications (R. Ansorge and W. Tornig, Eds.), Lecture Notes in Math., vol. 679, SpringerVerlag, Berlin and New York, 1978, pp. 8591. MR 515572 (81i:65066)
 [11]
 M. H. Protter & H. P. Weinberger, Maximum Principles in Differential Equations, PrenticeHall, Englewood Cliffs, N.J., 1967. MR 0219861 (36:2935)
 [12]
 S. A. Pruess, "Solving linear boundary value problems by approximating the coefficients," Math. Comp., v. 27, 1973, pp. 551561. MR 0371100 (51:7321)
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 M. E. Rose, "Weakelement approximations to elliptic differential equations," Numer. Math., v. 24, 1975, pp. 185204. MR 0411206 (53:14944)
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 D. R. Smith, "The multivariable method in singular perturbation analysis," SIAM Rev., v. 17, 1975, pp. 221273. MR 0361331 (50:13776)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198106163610
PII:
S 00255718(1981)06163610
Article copyright:
© Copyright 1981 American Mathematical Society
