1549301742-The_Theory_of_Difference_Schemes__Samarskii

The surnmarized approximation n1ethod^629

13. Alternating direction methods as additive sche1nes. The starting point
    for special investigations is the alternating direction method developed in
    Section l:

(60)

T

As can readily be observed, this system of equations is equivalent to

yi+l/2 - yi+l/4

------= 0.5A 1 yi+^1!^2 ,

T

( 61)

T

T

relating to locally one-dimensional schemes. Both intermediate values

yi+^114 and ~l+^3 /^4 can be eliminated in the usual way without any difficul-

ties. It is straightforward to verify that scheme (61) generates a summarized

approxi1nation such that

The alternating direction sche1ne ascribed to Duglas and Rachford

and associated with the difference operator Aa y = Yxaxa comes second:

(62)

T

Si1nple algebra gives the residuals