1549301742-The_Theory_of_Difference_Schemes__Samarskii

684 Methods for Solving Grid Equations

This implies that

meaning B = B*. By the same token,

It seems clear from the equation

given Yo EH,

that the value Yk+l remains as yet unknown and can be found from the

equation

(E+wRi)(E+wR2)Yk+1 = Fk

with the right-hand side Fk = Byk -Tk+ 1 (Ayk -f).

As a matter of experience, by solving successively two equations

(24)

with upper and lower triangle matrices we get a final result. So, the title and
origin of the alternative-triangle inethod (ATM) owe a debt to scheme
(23) with operator (22) that can acid interest and help in understanding of
how one could progress in this direction. Before going further, we need to
know the values of parameters / 1 and / 2.

Le1nma 2 Let operators R, R 1 and R 2 be in line with the conditions

R = R1 + R2, R; = R1 and

(25)

(26)

R= R*, R> 8E,

~

R1Ro ~ < -- 4 R,

Then the estimates are valid

(27)

8 > 0 J

~ > 0.