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Article Navigation > Applied Mathematics and Mechanics > 1999 > 20(1): 71-78

Cheng Aijie. Improvement on Stability and Convergence of A. D. I. Schemes[J]. Applied Mathematics and Mechanics, 1999, 20(1): 71-78.

Citation:

Cheng Aijie. Improvement on Stability and Convergence of A. D. I. Schemes[J]. Applied Mathematics and Mechanics, 1999, 20(1): 71-78.

Cheng Aijie. Improvement on Stability and Convergence of A. D. I. Schemes[J]. Applied Mathematics and Mechanics, 1999, 20(1): 71-78.

Citation:

Cheng Aijie. Improvement on Stability and Convergence of A. D. I. Schemes[J]. Applied Mathematics and Mechanics, 1999, 20(1): 71-78.

Improvement on Stability and Convergence of A. D. I. Schemes

Cheng Aijie

Department of Mathematics, Shandong University, Jinan 250100, P R China

Received Date: 1996-09-09
Rev Recd Date: 1998-10-06
Publish Date: 1999-01-15

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 ∂_u/∂_t-∂/∂_x[a(x,y,t)∂_u/∂x-∂/∂_y^o[nb(x,y,t)∂u/∂_y^cs]B=f 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 "equiverlence 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 O(Δt²＋h²) in discrete H¹ norm. This implies essential improvement of existing conclusions.
- P-R scheme,
- Douglas scheme,
- parabolic partial differential equation,
- variable coefficient,
- H¹ energy estimating method

References

[1]	Peaceman D W,Rachford H H.The numerical solution of parabolic and elliptic differential equations[J].J SIAM,1955,3(1):28～41
[2]	Douglas J Jr.Alternating direction methods for three space variables[J].Numer Math,1962,4(1):41～63
[3]	Lees M.Apriori estimates for the solutions of difference approximation to parabolic partial differential equations[J].Duke Math J,1960,27(3):297～311
[4]	Lees M.Alternating direction and semi-explicit difference method for parabolic partial differential
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[6]	Douglas J Jr,Gunn J E.A general formulation of alternating direction method[J].Numer Math,1964,6(5):428～453
[7]	Lees M.Comment on[5][J].Math Review,1966,31(1):159～160
[8]	Marchuk G I.Methods of Numerical Mathematics[M].Springer-Verlag,1981
[9]	Yanenko N N.The Method of Fractional Steps[M].Spring-Verlag,197.

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