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Unformatted text preview: J. Soc. INDUST. APPL. MATH. Vol. 3, No. 1, March, 1955 Printed in U.S.A. THE NUMERICAL SOLUTION OF PARABOLIC AND ELLIPTIC DIFFERENTIAL EQUATIONS* D. W. PEACEMAN AND H. H. RACHFORD, JR. Introduction. Numerical approximations to solutions of the heat flow equationintwo spacedimensionsmay be obtainedby thestepwisesolution of an associated difference equation. Two types of difference equations have previously been studied: (1)explicit difference equations, which are simple to solve, but which require an uneconomically large number of time steps oflimited size, and (2)implicit difference equations, which do not limit the time step but which require at each time step the solution by iterationoflargesetsofsimultaneous equations. In this paper, an alternatingdirection implicit procedure is presented that requires the linebyline solution of small sets of simultaneous equa tionsthat can be solvedby a direct,noniterativemethod. Analysis ofthe procedureshows ittobe stableforany sizetime stepand torequiremuch lesswork than other methods that have been studied. As a practicaltest, thenew procedurewas used tosolvetheheatflowequationwith boundary conditionsforwhich the formal solutionisknown; the two solutions were in good agreement. Inaddition,thealternatingdirectionimplicitmethod isapplicabletothe iterativesolution oftwodimensional steadystateproblems. Ina practical test,rapid convergence forthe solution of Laplaces equation in a square was obtained by using a suitable set of iteration parameters which were easilycalculated. An analysisispresentedthatshowsthemethodtorequire about (2 log N)/N as many calculations as the best previously known iterative procedureforsolvingLaplacesequation, where N isthenumber ofpointsforwhich the solutioniscomputed. In the firstpart ofthe paper, the numerical solution ofunsteadystate problems in two dimensions is discussed. For illustrative purposes, only thesimplesttype ofproblemisconsidered,thatofunsteadystateheatflow in a square. In the second part of the paper, the numerical solution of steadystate problems in two dimensions is discussed. Again, only the simplest type problem is considered, namely, the solution of Laplaces equation ina square. Inboth parts ofthe paper, the analyses willbe per formed forspecialboundary conditions.Theseanalysescan,inmany cases, be extended in a straightforward manner to problems having different boundaryconditions. *Received by the editors October 18, 1954. f Presented to the American Mathematical Society, August 31, 1954. 28 NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 29 The theoretical aspects of the procedures discussed here are treated in greaterdetailina companion paper by Douglas [5]....
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