1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternative-triangular inethod

under the inner product structure

Ni N2-1

(1t, vL = L L 11i,i2 vi,i2 h1 h2.

i1=l i2=l

The inner product (1t, v] 2 can be introduced in a similar way.

689

Knowing b and ~, it is not difficult to determine the values of 17, / 1 ,

/ 2 and~ and, after this, to estimate the number of the iterations required

in such n1atters.

Available data processing is a special starting procedure in the further

comparison of three rnethods in line with established priorities by having

recourse to the model problem (35) that we have set up on a square grid

h 1 = h 2 = h in the square G with the unit sides (! 1 = [ 2 = 1). Plain

calculations show that

. 7r h

1. = sm^2 - ,
   2

2 y'rj 7r h

~ = ~ 2 y'rj = 2 sin -.

l+J17 2

For srnall value.s " 2 h ~ 1 we might have

0.29 2 2 9
n (c) ~ --In - = - for c = 2 e-^10 ~ 10-^4.
0 v'h c Jh
The first criterion among others is the number of the iterations in the
following three schemes:

h SIS SCP ATM

1/10^200 32 9

1/50 5000 160 21

1/100 20000 320 29

We present below the algorithm of finding the ( k + 1 )th iteration from

the equation

(38)

k k k k k

F = By - Tk+l (A y - <p) = By + Tk+l (Av + !) ,