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In my Numerical Methods textbook (Applied Numerical Analysis, 7ed, Gerald and Wheatley) the authors derive two equations for the ADI method to be used in an iteration scheme. For row-wise traversions, they get

[tex]uO^{(k+1)}=uO^{(k)}+\rho(uL-2uO+uR)^{(k+1)}+\rho(uA-2uO+uB)^{(k)}[/tex]

and for column-wise traversions

[tex]uO^{(k+2)}=uO^{(k+1)}+\rho(uA-2uO+uB)^{(k+2)}+\rho(uL-2uO+uR)^{(k+1)}[/tex]

Where u represents the nodes, A, B, L, R are above, below, left and right respectively, O is the node in the centre (current), k represents the iteration and rho is an acceleration factor.

I understand that we alternate between these equations for successive iterations (hence the name ) but what I don't get is that it seems to me that the value we're trying to calculate is dependent on itself, e.g how do we determine [tex]uO^{(k+1)}[/tex] if we don't yet have [tex](uL-2uO+uR)^{(k+1)}[/tex] ?

Any insight will be greatly appreciated!

phyz

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# The Alternating Direction Implicit Method in 2-D

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