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Author Country (or Countries)

South Africa

Abstract

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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