Alternating-direction implicit method: Difference between revisions

Revision as of 17:05, 22 January 2010

Revision as of 17:09, 22 January 2010

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\left(\delta_x^2 u_{ij}^{n+1/2}+\delta_y^2 u_{ij}^{n+1}\right).</math>

The systemssystem of equations involved areis [[symmetric matrix|symmetric]] and tridiagonal (banded with bandwidth 3), and thusis cheaptypically tosolved solve byusing [[Choleskitridiagonal matrix decompositionalgorithm]].

It can be shown that this method is unconditionally stable. There are more refined ADI methods such as the methods of Douglas<ref>{{Citation | last1=Douglas Jr. | first1=Jim | title=Alternating direction methods for three space variables | doi=10.1007/BF01386295 | year=1962 | journal=Numerische Mathematik | issn=0029-599X | volume=4 | pages=41–63}}.</ref>, or the f-factor method<ref>{{Citation | last1=Chang | first1=M. J. | last2=Chow | first2=L. C. | last3=Chang | first3=W. S. | title=Improved alternating-direction implicit method for solving transient three-dimensional heat diffusion problems | doi=10.1080/10407799108944957 | year=1991 | journal=Numerical Heat Transfer, Part B: Fundamentals | issn=1040-7790 | volume=19 | issue=1 | pages=69–84}}.</ref> which can be used for three or more dimensions.