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Mathematics of Computation

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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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Article copyright: © Copyright 1981 American Mathematical Society