Improvement on stability and convergence of A. D. I. schemes |
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Authors: | Cheng Aijie |
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Institution: | Department of Mathematics, Shandong University, Jinan 250100, P. R. China |
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Abstract: | Alternating direction implicit (A. D. I.) schemes have been proved valuable in the approximation of the solutions of parabolic
partial differential equations in multi-dimensional space. Consider equations in the form
Two A. D. I. schemes, Peaceman-Rachford scheme and Douglas scheme will be studied. In the literature, stability and convergence
have been analysed with Fourier Method, which cannot be extended beyond the model problem with constant coefficients. Additionally,
L2 energy method has been introduced to analyse the case of non-constant coefficients, however, the conclusions are too weak
and incomplete because of the so-called ”equivalence between L2 norm and H1 semi-norm”. In this paper, we try to improve these conclusions by H1 energy estimating method. The principal results are that both of the two A. D. I. schemes are absolutely stable and converge
to the exact solution with error estimations 0(Δt2+h2) in discrete H1 norm. This implies essential improvement of existing conclusions.
Project supported by the National Natural Science Foundation of China |
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Keywords: | P-R scheme Douglas scheme parabolic partial differential equation variable coefficient H1 energy estimating method stability and convergence |
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