A note on a generalisation of a method of Douglas

Author:
Graeme Fairweather

Journal:
Math. Comp. **23** (1969), 407-409

MSC:
Primary 65.68

DOI:
https://doi.org/10.1090/S0025-5718-1969-0243756-4

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 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]**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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DOI:
https://doi.org/10.1090/S0025-5718-1969-0243756-4

Article copyright:
© Copyright 1969
American Mathematical Society