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Analysis of an alternating direction method applied to singularly perturbed reaction-diffusion problems. (English) Zbl 1195.65119
Summary: We present an analysis of an alternating direction implicit (ADI) scheme for a linear, singularly perturbed reaction-diffusion equation. By providing an expression for the error that separates the temporal and spatial components, we can use existing results for steady-state problems to give a succinct analysis for the time-dependent problem, and that generalizes for various layer-adapted meshes. We report the results of numerical experiments that support the theoretical findings. In addition, we provide a numerical comparison between the ADI and Euler techniques, as well details of the computational advantage gained by parallelizing the algorithm.

MSC:
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35B25 Singular perturbations in context of PDEs
65F10 Iterative numerical methods for linear systems
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
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