1549301742-The_Theory_of_Difference_Schemes__Samarskii

656 Methods for Solving Grid Equations

Also, it seems clear from (9) that

where .'h+ 1 is the transition operator from the layer k to the layer k + 1.

Having completed the elin1ination of zk, zk~l, ... , z 1 , we find fork= n - 1

that

where Tn is the resolving operator of scheme (9). This serves to motivate

the estimates

From such reasoning it seems clear that the condition of termination is

ensured if qn < c;, thereby reducing the question of convergence of the

iterations to the norm estirnation of the resolving operator Tn.

Scheme (3) generates an exact. approxin1ation on a solution ·u of the

equation Att = f for any operators { B 11 } and any choice of the parameters

{ Tk+l}, but the quantity q 11 depends on {En} and { Tk+l} both. Son1e

consensus of opinion here is that {En} and { Tk+l} should be so chosen as

to minimize the norm //Tn//D = qn of the resolving operator T 11 of scheme

(3) and to minimize the total number of arithmetic operations which will

be needed for recovering the value Yk+l from the equation

with a known value Yk+l.

In accordance with what has been said above, any iteration scheme

(3) can be treated as a two-layer scheme being used for solving the nonsta-

tionary problem

where the para1neter Tk+l regards to one possible step 111 a nonreal ti1ne

tk+l = :z:+=\ Tm. The main differences between iteration schemes and
available schemes for nonstationary problems are:

• the iteration scheme (3) approximates exactly equation (1), since
  a solution u to equation (1) satisfies equation (3) for any Bk and
  Tk+l;