Alternating-direction implicit method: Difference between revisions

Revision as of 16:29, 16 May 2017

Revision as of 16:29, 16 May 2017

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

where:

with:

: <math>\Delta \equiv \Delta x = \Delta y</math>

whereand <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 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]]).