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A note on a generalisation of a method of Douglas

Author: Graeme Fairweather
Journal: Math. Comp. 23 (1969), 407-409
MSC: Primary 65.68
MathSciNet review: 0243756
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Abstract: In this note, the high-order correct method of Douglas [1] for the diffusion equation in one space variable is extended to $ q \leqq 3$ space variables. The resulting difference equations are then solved using the A. D. I. technique of Douglas and Gunn [3]. When $ q = 2$, this method is equivalent to that of Mitchell and Fairweather [5] while $ q = 3$ provides a method which is similar to Samarskiï's method [6] and of higher accuracy than that of Douglas [2]. When the proposed methods are used to solve the diffusion equation with timeindependent boundary conditions, they have the advantage that no boundary modification (see [4]) is required to maintain accuracy.

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  • [1] J. Douglas, Jr., ``The solution of the diffusion equation by a high order correct difference equation,'' J. Math. Phys., v. 35, 1956, pp. 145-151. MR 19, 884. MR 0090875 (19:884f)
  • [2] J. Douglas, Jr., ``Alternating direction methods for three space variables,'' Numer. Math., v. 4, 1962, pp. 41-63. MR 24 #B2122. MR 0136083 (24:B2122)
  • [3] J. Douglas, Jr. & J. E. Gunn, ``A general formulation of alternating direction methods. I. Parabolic and hyperbolic problems,'' Numer. Math., v. 6, 1964, pp. 428-453. MR 31 #894. MR 0176622 (31:894)
  • [4] G. Fairweather & A. R. Mitchell, ``A new computational procedure for A.D.I, methods,'' SIAM J. Numer. Anal., v. 4, 1967, pp. 163-170. MR 36 #1116. MR 0218027 (36:1116)
  • [5] A. R. Mitchell & G. Fairweather, ``Improved forms of the alternating direction methods of Douglas, Peaceman and Rachford for solving parabolic and elliptic equations,'' Numer. Math., v. 6, 1964, pp. 285-292. MR 30 #4391. MR 0174184 (30:4391)
  • [6] A. A. Samarskiï, ``A difference scheme for increasing the order of accuracy for the heat equation in several variables,'' Z. Vyčisl. Mat. i Mat. Fiz., v. 4, 1964, pp. 161-165. MR 31 #4197. MR 0179960 (31:4197)

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Article copyright: © Copyright 1969 American Mathematical Society

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