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In [[numerical analysis]], the '''Alternating Direction Implicit (ADI) method''' is a [[finite difference]] method for solving [[Parabolic partial differential equation|parabolic]] and [[Elliptic partial differential equation|elliptic]] partial differential equations.<ref>{{Citation | doi=10.1137/0103003 | last1=Peaceman | first1=D. W. | last2=Rachford Jr. | first2=H. H. | 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 | issue=1 | 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. It is an example of an <i>operator splitting</i> method.<ref>*{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 20.3.3. Operator Splitting Methods Generally | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=1052}}
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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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