1549301742-The_Theory_of_Difference_Schemes__Samarskii

634 Economical Difference Schemes for Multidhnensional Problems

With the basic tools in hand, we proceed to carry out the accurate

account of the approximation error z(a) = z]+a/^2 = yj+a/^2 - u(x, tj+c'/ 2 )

for scheme (75)-(77), where v. is a solution of problem (63)-(65) and y is a

solution of problem (75)-(77), by inserting the value Y(a) = z(a) + ttj+a/^2

in equation (75). As a final result we obtain

(78) for

for t = 0.5r,

z(x, 0) = 0, Z (a) -zj+a/2_0 - - for XE/~, cx=l,2,

with the member

(79)

serving as one possible representation for the residual of equation (66) with

the number CY = 1, 2. The error of approxi1nation for LOS of the forn1 (75

)-(77) is viewed as a sum

(80)

Further progress in this area will be achieved by utilizing the fact that
sche111e ( 76) approxinrn.tes problen1 ( 63 )-( 65) in a smn111arized sense and
4J = O(r + lhl^2 ). Indeed, taking into account that

(^0). (^5) i . a ( ll 0 + lla - ) = (L Lr tl )j+(a-1)/2 + O(h2) a for
for x E w 1 * i,a ,
u- - - -^1 (fJ2·u)j+(o:--^1 Jn + O(r^2 )
tata - 4 [)t2^1
0
we find that 1/J a = 1/J a + 1/J:, where
0 - ~ ( - ~ a2v )j+(cr-1)/2
1/J o: - 2 Lull 2 [)t2 + fa ' c~=l,2,