Em [[análise numérica]], the '''Alternating Direction Implicit (ADI) method''' is a [[finite difference]] method for solving [[Parabolic partial differential equation|parabolic]], [[Hyperbolic partial differential equation|hyperbolic]] 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 | mr=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 ''operator splitting'' 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 numerically is the [[Crank–Nicolson method]]. This method results in a very complicated set of equations in multiple dimensions, which are costly to solve. The advantage of the ADI method is that the equations that have to be solved in each step have a simpler structure and can be solved efficiently with the [[tridiagonal matrix algorithm]].
== OThe métodomethod ==
Considere a equação linear da difusão do calor em duas dimensões,
