Gradedmesh difference schemes for singularly perturbed twopoint boundary value problems
Author:
Eugene C. Gartland
Journal:
Math. Comp. 51 (1988), 631657
MSC:
Primary 65L10; Secondary 34E15
MathSciNet review:
935072
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Abstract: This paper is concerned with the numerical approximation by compact finitedifference schemes of differential operators of the form without turning points. The stability of combined with various auxiliary conditions is discussed, and a representation result for solutions of problems involving it is proven. This representation decomposes the solution into a smooth outer component plus a decaying exponential layer term along the lines of the Method of Multiple Scales. The stability of compact difference analogues of is studied, and a stability result is proven which generalizes earlier work. This result encompasses, for example, discretizations of secondorder problems that fail to possess a maximum principle. It allows for standard polynomialbased differences in outer regions (away from boundary layers) with uniform meshes, even though such schemes admit oscillatory solutions. A family of finitedifference schemes based on an exponentially graded mesh and local polynomial basis functions is discussed. These schemes can be constructed to have arbitrarily high uniform order of convergence. To achieve a scheme of order , roughly K times as many points are distributed inside the layer as outside. The high order is achieved by using extra local evaluations of the coefficient functions and source term of the problem. A rigorous discretization error analysis of these schemes, using the established stability and representation results, is given. Numerical results exhibiting the performance of these schemes are presented and generalizations of the results in the paper are discussed.
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 U. Ascher & R. Weiss, "Collocation for singular perturbation problems I: First order systems with constant coefficients," SIAM J. Numer. Anal., v. 20, 1983, pp. 537557. MR 701095 (85a:65113)
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 U. Ascher & R. Weiss, "Collocation for singular perturbation problems II: Linear first order systems without turning points," Math. Comp., v. 43, 1984, pp. 157187. MR 744929 (86g:65138a)
 [3]
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 A. E. Berger, J. M. Solomon & M. Ciment, "Uniformly accurate difference methods for a singular perturbation problem," in Boundary and Interior Layers Computational and Asymptotic Methods, Proc. BAIL I Conf. (J. J. H. Miller, ed.), Boole Press, Dublin, 1980. MR 589348 (82e:65083)
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 E. C. Gartland, Jr., Strong Stability and a Rrepresentation Result for a Singular Perturbation Problem, Technical Report AMS 871, January, 1987, Department of Mathematics, Southern Methodist University.
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 E. C. Gartland, Jr., "Uniform highorder difference schemes for a singularly perturbed twopoint boundary value problem," Math. Comp., v. 48, 1987, pp. 551564. MR 878690 (89a:65116)
 [10]
 E. C. Gartland, Jr., "Strong stability of compact discrete boundary value problems via exact discretizations," SIAM J. Numer. Anal., v. 25, 1988, pp. 111123. MR 923929 (89a:65127)
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 H.O. Kreiss, "Difference approximations for boundary and eigenvalue problems for ordinary differential equations," Math. Comp., v. 26, 1972, pp. 605624. MR 0373296 (51:9496)
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 P. A. Markowich & C. A. Ringhofer, "Collocation methods for boundary value problems on 'long' intervals," Math. Comp., v. 40, 1983, pp. 123150. MR 679437 (84d:65053)
 [17]
 K. Niederdrenk & H. Yserentant, "Die gleichmässige Stabilität singulär gestörter diskreter und kontinuierlicher Randwertprobleme," Numer. Math., v. 41, 1983, pp. 223253. MR 703123 (84j:65049)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809350721
PII:
S 00255718(1988)09350721
Keywords:
Singular perturbations,
boundary value problems,
finite differences,
stability,
uniform methods,
graded meshes
Article copyright:
© Copyright 1988
American Mathematical Society
