Stability and convergence of the PeacemanRachford ADI method for initialboundary value problems
Authors:
W. H. Hundsdorfer and J. G. Verwer
Journal:
Math. Comp. 53 (1989), 81101
MSC:
Primary 65N40; Secondary 65M20
MathSciNet review:
969489
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Abstract: In this paper an analysis will be presented for the ADI (alternating direction implicit) method of Peaceman and Rachford applied to initialboundary value problems for partial differential equations in two space dimensions. We shall use the method of lines approach. Motivated by developments in the field of stiff nonlinear ordinary differential equations, our analysis will focus on problems where the semidiscrete system, obtained after discretization in space, satisfies a onesided Lipschitz condition with a constant independent of the grid spacing. For such problems, unconditional stability and convergence results will be derived.
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 K. Dekker & J. G. Verwer, Stability of RungeKutta Methods for Stiff Nonlinear Differential Equations, NorthHolland, Amsterdam, 1984. MR 774402 (86g:65003)
 [2]
 J. Douglas, Jr. & J. E. Gunn, "A general formulation of alternating direction methods, Part I: Parabolic and hyperbolic problems," Numer. Math., v. 6, 1964, pp. 428453. MR 0176622 (31:894)
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 J. Douglas, Jr. & H. H. Rachford, Jr., "On the numerical solution of heat conduction problems in two and three space variables," Trans. Amer. Math. Soc., v. 82, 1956, pp. 421439. MR 0084194 (18:827f)
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 P. J. van der Houwen & J. G. Verwer, "Onestep splitting methods for semidiscrete parabolic equations," Computing, v. 22, 1979, pp. 291309. MR 620058 (83e:65148)
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 P. J. van der Houwen & B. P. Sommeijer, Improving the Stability of PredictorCorrector Methods by Residue Smoothing, Report NMR8707, Centre for Mathematics and Computer Science, Amsterdam, 1987.
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 H. F. B. M. Kraaijevanger, "Bconvergence of the implicit midpoint rule and the trapezoidal rule," BIT, v. 25, 1985, pp. 652666. MR 811280 (87c:65096)
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 R. D. Richtmeyer & K. W. Morton, Difference Methods for InitialValue Problems, Interscience, New York, 1967.
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 J. M. SanzSerna, J. G. Verwer & W. H. Hundsdorfer, "Convergence and order reduction of RungeKutta schemes applied to evolutionary partial differential equations," Numer. Math., v. 50, 1987, pp. 405418. MR 875165 (88f:65146)
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 J. M. SanzSerna & J. G. Verwer, "Stability and convergence in the stiff ODE/PDE interface," Appl. Numer. Math., v. 5, 1989, pp. 117132. MR 979551 (90c:65126)
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 B. P. Sommeijer, P. J. van der Houwen & J. G. Verwer, "On the treatment of timedependent boundary conditions in splitting methods for parabolic differential equations," Internat. J. Numer. Math. Engrg., v. 17, 1981, pp. 335346. MR 608685 (83b:65093)
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 J. G. Verwer, "Contractivity of locally onedimensional splitting methods," Numer. Math., v. 44, 1984, pp. 247259. MR 753957 (85j:65038)
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 J. G. Verwer & J. M. SanzSerna, "Convergence of method of lines approximations to partial differential equations," Computing, v. 33, 1984, pp. 297313. MR 773930 (86k:65085)
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 J. G. Verwer & H. B. de Vries, "Global extrapolation of a first order splitting method," SIAM J. Sci. Statist. Comput., v. 6, 1985, pp. 771780. MR 791198
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 J. G. Verwer, "Convergence and order reduction of diagonally implicit RungeKutta schemes in the method of lines," Numerical Analysis (D. F. Griffiths and G. A. Watson, eds.), Pitman Research Notes in Mathematics Series, vol. 140, 1986, pp. 220237. MR 873112 (88f:65158)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198909694897
PII:
S 00255718(1989)09694897
Keywords:
Numerical analysis,
timedependent PDE's,
alternating direction implicit methods,
PeacemanRachford method,
method of lines,
stability,
error bounds
Article copyright:
© Copyright 1989
American Mathematical Society
