1549301742-The_Theory_of_Difference_Schemes__Samarskii

Econmnical factorized schemes^587

e1Tor the following conditions:

z=O forxE11i, tEw 7 , z(x,0)=0 forxEw1i,

where

(60)

1/1 0 is the error of approximation of the primary scheme (56) and

V = U = Ut = Q ( T).

Since B = E + 2T^2 Qp > E, Theorems 6 and 9 in Chapter 6, Section 3 are

still valid for scheme ( 59), on account of which the error of approximation

v to the second initial condition can be most readily evaluated in the norm

ffvffn, where

ffvll~ = (Dv,v) = T^2 (Rv,v)+T^3 (Qpv,v) = 0(T^4 ).

ffvffn = 0(T^2 ), since v = O(T).

From (60) it is easily seen that 1/J = 0( T^2 + fhf^2 ). The smoothness proper-
ties due to which we would have 1/J = 0(T^2 + fhf^2 ) and ffvffn = 0(T^2 ) will
become more stronger in connection with increasing the number of obser-
vations p. These restrictions can be relaxed with the aid of the chain of the
inequalities

p

2 T (1/J, Zo) t = 2 T^3 (Qpv, Zo) t = 2 T^3 (Qp(2lv, Zo) t + 2 T^3 ~ L..., Ts-^2 (Q~s)V, Zo) t

s=3

p p

< Tffzt 112+T5flQ~2)vff2+ L Ts(Q~s)z,zt)+ L Ts+2(Q~slv,v)

s=3 ~=3