1549301742-The_Theory_of_Difference_Schemes__Samarskii

(jair2018) #1
Homogeneous difference schemes for the heat conduction 473

By substituting estimates (30) and (31) into inequality (40) and, in turn,
0
estimates (33) and (34) into a sin1ilar inequality for II A-^1 'ljJ II and then
applying estimates (35), (38), (39) and the inequality II z II < II z* II+ II v II
t<,J the resulting expressions we establish the convergence of scheme (7) in
the class of discontinuous coefficients.

Let k(x) possess a discontinuity of the first kind for
x = ~ and conditions (23), (24), (30) and (31) hold
simultaneously. Then for CJ > CJ 0 , CJ > 0, scheme (7)
con verges in the norm of the grid space L 2 with the rate
O(h^2 + rm+^0 ), while the best schenie with coefficient
(32) does the same with the rate O(h^2 +rm+^0 ); m 0 = 2
for CJ= 0.5 and 1120 = 1 for CJ f:- 0.5.

A priori estimates obtained in Chapter 6, Section 2 provides the suf-
ficient background for a· uniform estimate of accuracy in the norm of the
grid space C such as

j
+I: rll~{llA-1 for CJ>CJ 0 ,
j'=l

where II z II~ = (;l_z, z) = (a, (z;;)~] and the inequa.lities (see Chapter 2,
Section 3) are taken into account:


In conformity with Chapter 2, Section 4 we deduce that
Free download pdf