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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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), 79-94
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: $ \varepsilon {u_{xx}} + b(x){u_x} = f(x)$ for $ 0 < x < 1,b > 0$, b and f smooth, $ \varepsilon $ in (0, 1], and $ u(0)$ and $ u(1)$ given, is uniformly second-order accurate (i.e., the maximum of the errors at the grid points is bounded by $ C{h^2}$ with the constant C independent of h and $ \varepsilon $). This scheme was derived by El-Mistikawy and Werle by a $ {C^1}$ 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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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1981-0616361-0
PII: S 0025-5718(1981)0616361-0
Article copyright: © Copyright 1981 American Mathematical Society