Alternating-direction implicit method: Difference between revisions

Revision as of 15:43, 6 October 2013

Revision as of 07:21, 20 October 2013

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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 direct solution of the [[system of linear equations]] quite costly (although efficient approximate solutions exist, for example use of the [[conjugate gradient method]] preconditioned with [[incomplete Cholesky factorization]]).

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 implicitlyexplicitly,

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