Abstract. In this paper a singularly perturbed reaction-diffusion partial differential equation in two space dimensions is examined. By means of an appropriate decomposition, we describe the asymptotic behaviour of the solution of problems of this kind. A central finite difference scheme is constructed for this problem which involves an appropriate Shishkin mesh. We prove that the numerical approximations are almost second order uniformly convergent (in the maximum norm) with respect to the singular perturbation parameter. Some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method.
In this work we construct and analyse some finite difference schemes used to solve a class of timedependent one-dimensional convection-diffusion problems, which present only regular layers in their solution. We use the implicit Euler or the Crank-Nicolson method to discretize the time variable and a HODIE finite difference scheme, defined on a piecewise uniform Shishkin mesh, to discretize the spatial variable. In both cases we prove that the numerical method is uniformly convergent with respect to the diffusion parameter, having order near two in space and order one or 3/2, depending on the method used, in time. We show some numerical examples according to the theoretical results, in the case of using the Euler implicit method, and a better numerical behaviour than the predicted theoretically, showing order two in time and order N −2 log 2 N in space, if the Crank-Nicolson scheme is used to discretize the time variable. Finally, we construct a numerical algorithm by combining a third order A-stable SDIRK with two stages and a third order HODIE difference scheme, showing its uniformly convergent behaviour, reaching order three, up to a logarithmic factor. c (Year) John Wiley & Sons, Inc.
In this paper we develop a numerical method for two-dimensional time-dependent reaction-diffusion problems. This method, which can immediately be generalized to higher dimensions, is shown to be uniformly convergent with respect to the diffusion parameter.
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