Alternating direction implicit method क्या है?

An alternating direction implicit method (Peaceman and Rachford, 1955; Douglas and Rachford, 1956) splits a multi-dimensional operator $L = A_0 + A_1 + A_2$, with $A_1$ and $A_2$ acting along one coordinate each and $A_0$ the mixed-derivative part, and treats one direction implicitly at a time. The Douglas scheme is $Y_0 = U^n + \Delta\tau LU^n$, $Y_j = Y_{j-1} + \theta\Delta\tau A_j(Y_j - U^n)$ for $j = 1, 2$, $U^{n+1} = Y_2$.