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 | idmr={{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 system of equations involved is [[symmetric matrix|symmetric]] and tridiagonal (banded with bandwidth 3), and is typically solved using [[tridiagonal matrix algorithm]].
It can be shown that this method is unconditionally stable and second order in time and space.<ref>{{Citation | last1=Douglas, Jr. | first1=J. | title=On the numerical integration of u<sub>xx</sub>+ u<sub>yy</sub>= u<sub>tt</sub> by implicit methods | idmr={{MathSciNet | id = 0071875}} | year=1955 | journal=Journal of the Society of Industrial and Applied Mathematics | volume=3 | pages=42–65}}.
</ref>. 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 | issue=1 | 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.
