1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternating direction 111ethod 561

where Bex= E - (}ex T Aex and Cex = E + (1-(}ex) T Aex, Cl'= 1, 2.

Having multiplied the first equation ( 48) by 1-() 1 and the second one

by () 1 and having added the final result, we find with the aid of the relation

(1-(} 1 )B 1 + (} 1 C'i = E tile intennediate value

( 40)

Substituting this expression into the first equation ( 48) yields

or

(50)

By observing that

h2 + h2

( 1 - (}! - (}?) T = I 2

• 12

this provides enough reason to rewrite the preciding sche1ne as

( 51)

(52) y(x, 0) = Ua(x) for x E wh, y" = μ" for x E 'Yh ,

where

Simple algebra gives

(53)

thereby justifying that schen1e (.51)-(52) generates an approxi111ation of