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A Robust Finite Difference Method for Two-Parameter Parabolic Convection-Diffusion Problems |
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PP: 2877-2883 |
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Author(s) |
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Justin B. Munyakazi,
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Abstract |
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| We consider a class of singularly perturbed parabolic differential equations with two small parameters affecting the
derivatives. The solution to such problems typically has parabolic layers. We discretize the time variable by means of the classical
backward Euler method. At each time level a two-point boundary value problem is obtained. These problems are, in turn, discretized in
space on a uniform mesh following the nonstandard methodology of Mickens. We prove that the underlying discrete operator satisfies
a minimum principle. We use this result in the error analysis. We show that the method is uniformly convergent with respect to the
perturbation parameters. This is contradictory with the assertion [G.I. Shishkin, A difference scheme for a singularly perturbed equation
of parabolic type with discontinuous initial condition, Soviet Math. Dokl. 37 (1988) 792-796] that parameter-uniform numerical
methods cannot be designed on a uniform mesh for problems whose solution exhibits parabolic layers. Finally we give numerical
results to attest the parameter-uniform convergence. Moreover, comparison with some existing methods in the literature proves the
competitiveness of our method. |
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