1549301742-The_Theory_of_Difference_Schemes__Samarskii

The s1unmarized approxiination 1nethod 619

A two-layer additive sche1ne can a.lways be written in the canonical fonn

yj+o:/m _ yJ+(o:-l)/m m

B + 2= Ao:r1if+f3/m

T f3=0

( 43) -- (/)] ra'

Ct=l,2,. .. ,m,

where B and A 1 , f3 are s01ne linear operators. It is :straightforward to verify

that all of the available econ01nical methods with canonical form ( 43) can

generate a summarized approximation.

Furthermore, let ·u( t) E Ho be an abstract function of the argutnent

t E [O, t 0 ] with the values in a normed space Ho and uh =Phu E Hh be

the projection of n onto H h,

is the residual for equation (43) with the number CY and v)+o:/m

TCY /m). Assuming this to be the case, the sun1

m

ij}(tlJ,) = 2= 00 ( vJ,)

CY=l

is of our initial concern.

By definition, the additive sche1ne ( 42) provides a summarized ap-

proxi1nation on a function u( t) E H 0 if

O:SJ:SJo max II 1/J(u^1 h ) II (2h) ~.^0 as T ----+^0 , h ----+^0 ,

where II · llc 2 h) is s01ne suitable nonn on the space Hh. In conformity with

Section 2 the additive scheme ( 42) is said to be econon1ical if the operator
(matrix) C is economical. That is to say, the work and storage necessary
in the numerical solution of the system of operator equations

m
( 44) 2= Co:f3 ~/+f3/m =~
/]=!