Alternating-direction implicit method: Difference between revisions

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In [[mathematics]], the '''alternating direction implicit (ADI) method''' is a [[finite difference]] method for solving parabolic and elliptic partial differential equations.<ref>{{Citation | last1=Peaceman, | first1=D. W. and| last2=Rachford, HJr. | first2=H., JrH., "| title=The numerical solution of parabolic and elliptic differential equations", | id={{MathSciNet | id = 0071874}} | year=1955 | journal=Journal of the Society for Industrial and Applied Mathematics | volume=3 | pages=28–41}}.</ref> It is most notably used to solve the problem of [[heat conduction]] or solving the [[diffusion equation]] in two or more dimensions.

SIAM J. 3 (1955), 28-41, [http://www.ams.org/mathscinet-getitem?mr=71874 MR71874]</ref>is a [[finite difference]] method for solving parabolic and elliptic partial differential equations. It is most notably used to solve the problem of [[heat conduction]] or solving the [[diffusion equation]] in two or more dimensions.

The traditional method for solving the heat conduction equation is the [[Crank–Nicolson method]]. This method can be quite costly. The advantage of the ADI method is that the equations that have to be solved in every iteration have a simpler structure and are thus easier to solve.

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A disadvantage of the Crank–Nicolson method is that the matrix in the above equation is [[band matrix|banded]] with a band width that is generally quite large. This makes the solution of the equation quite costly.

The idea behind the ADI method is to split the finite difference equations into two, one with the ''x''-derivative taken implicitly and the next with the ''y''-derivative taken explicitlyimplicitly,

: <math>{u_{ij}^{n+1/2}-u_{ij}^n\over \Delta t/2} =

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The systems of equations involved are [[symmetric matrix|symmetric]] and tridiagonal (banded with bandwidth 3), and thus cheap to solve by [[Choleski decomposition]].

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, JJr. "| first1=Jim | title=Alternating direction methods for three space variables," | doi=10.1007/BF01386295 | year=1962 | journal=Numerische Mathematik, Vol| issn=0029-599X | volume=4., pp| 41-63 (1962)pages=41–63}}.</ref>, or the f-factor method<ref>{{Citation | last1=Chang,Jefferies, | first1=M. J. et| allast2=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, VolPart 19,B: ppFundamentals 69| issn=1040-84,7790 (1991)| volume=19 | issue=1 | pages=69–84}}.</ref> which can be used for three or more dimensions.

== References ==