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 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****[2]**Jim Douglas Jr.,*Alternating direction methods for three space variables*, Numer. Math.**4**(1962), 41–63. MR**0136083****[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****[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****[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****[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**

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

Article copyright:
© Copyright 1969
American Mathematical Society