Alternating-direction implicit method: Difference between revisions

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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 method of [[Crank-NicolsonCrank–Nicolson method]]. This method can be quite costly. The advantage of the ADI method is implicit,that butthe hasequations anthat unaffordablehave stabilityto criterionbe solved in twoevery iteration have a simpler structure and are thus oreasier moreto dimensionssolve.

== The method ==

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= ( u_{xx} + u_{yy} ) \quad </math>

The implicit Crank-NicolsonCrank–Nicolson method produces the following finite difference equation:

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

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where <math>\delta_p^2</math> is the central difference operator for the ''p''-coordinate.

After performing a stability analysis, it can be shown that this method will be stable for any ''r''. because this method is implicit.

ButA problemdisadvantage isof the final matrix toCrank–Nicolson solvemethod is notthat tridiagonal,the butmatrix is banded. Depending onin the orderingabove (theequation numberingis of[[band thematrix|banded]] points),with thea bandwidthband ofwidth matrixthat canis begenerally veryquite large,. andThis this causesmakes the solution procedureof tothe beequation morequite difficultcostly.

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 implicitly,

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

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

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

The systems of equations involved are tri-diagonal ([[symmetric matrix|symmetric]] [[Bandand matrix|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>Douglas, J. "Alternating direction methods for three space variables," Numerische Mathematik, Vol 4., pp 41-63 (1962)</ref>, or the f-factor method<ref>Chang, M.J. et al. "Improved alternating-direction implicit method for solving transient three-dimensional heat diffusion problems", Numerical Heat Transfer, Vol 19, pp 69-84, (1991)</ref> which can be used for three or more dimensions.