A note on a generalisation of a method of Douglas
Author:
Graeme Fairweather
Journal:
Math. Comp. 23 (1969), 407409
MSC:
Primary 65.68
MathSciNet review:
0243756
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Abstract: In this note, the highorder correct method of Douglas [1] for the diffusion equation in one space variable is extended to space variables. The resulting difference equations are then solved using the A. D. I. technique of Douglas and Gunn [3]. When , this method is equivalent to that of Mitchell and Fairweather [5] while 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.
 [1]
Jim
Douglas Jr., The solution of the diffusion equation by a high order
correct difference equation, J. Math. and Phys. 35
(1956), 145–151. MR 0090875
(19,884f)
 [2]
Jim
Douglas Jr., Alternating direction methods for three space
variables, Numer. Math. 4 (1962), 41–63. MR 0136083
(24 #B2122)
 [3]
Jim
Douglas Jr. and James
E. Gunn, A general formulation of alternating direction methods. I.
Parabolic and hyperbolic problems, Numer. Math. 6
(1964), 428–453. MR 0176622
(31 #894)
 [4]
G.
Fairweather and A.
R. Mitchell, A new computational procedure for
𝐴.𝐷.𝐼. methods, SIAM J. Numer. Anal.
4 (1967), 163–170. MR 0218027
(36 #1116)
 [5]
A.
R. Mitchell and G.
Fairweather, Improved forms of the alternating direction methods of
Douglas, Peaceman, and Rachford for solving parabolic and elliptic
equations, Numer. Math. 6 (1964), 285–292. MR 0174184
(30 #4391)
 [6]
A.
A. Samarskiĭ, A difference scheme for increasing the order
of accuracy for the heat equation in several variables, Ž.
Vyčisl. Mat. i Mat. Fiz. 4 (1964), 161–165
(Russian). MR
0179960 (31 #4197)
 [1]
 J. Douglas, Jr., ``The solution of the diffusion equation by a high order correct difference equation,'' J. Math. Phys., v. 35, 1956, pp. 145151. MR 19, 884. MR 0090875 (19:884f)
 [2]
 J. Douglas, Jr., ``Alternating direction methods for three space variables,'' Numer. Math., v. 4, 1962, pp. 4163. 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. 428453. 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. 163170. 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. 285292. 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. 161165. MR 31 #4197. MR 0179960 (31:4197)
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DOI:
http://dx.doi.org/10.1090/S00255718196902437564
PII:
S 00255718(1969)02437564
Article copyright:
© Copyright 1969 American Mathematical Society
