Alternating-direction implicit method: Difference between revisions

Revision as of 15:40, 13 July 2011

Revision as of 23:44, 17 July 2011

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where <math>\delta_p</math> is the central difference operator for the ''p''-coordinate. After performing a [[Von Neumann stability analysis|stability analysis]], it can be shown that this method will be stable for any <math>\Delta t</math>.

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 thedirect solution of the equation[[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 implicitly,