1549301742-The_Theory_of_Difference_Schemes__Samarskii

Convergence and accuracy of hon1ogeneous conservative schemes 163

It seems worthwhile to focus the reader's attention on the best scherne

(15) from Section 2 for which the chain of the relations occurs:

0 1

o-2-- = J __ d_s __ = .1--c_ls __

a k(:t:n+i +sh) k(x 11 +sh)

n+l -1 Q

e 1

J

ds J ds

= k(x 71 +sh) + k(x 71 +sh)

o e

e

= j (-k

1

+ (s - 8) h

1

(-k

1

) + O(h

2

)) ds

left left

0

l

+ j (k

1

+ (s - 8) h^1 (_kl).. + O(h^2 )) ds

nght 11ght

e

8 1 - 8

= -+ +O(h),

kleft kright

0
so that 7] 71 + 1 = 1771 + 1 = O(h). For all other schernes 1Jn+i = 0(1).

In the estimation of 1/;* it is necessary to distinguish two possibilities
of interest: 8 < 0.5 and 8 > 0.5.

(1) Let 8 < 0.5. In every such case 1/J; = O(h^2 ) for i f:- n, while

0 0

1/;~ = 0(1) and only for the best scheme with <p, = <p 1 and d; = d;

0 1/2

incorporated 1/J; = J q(xi +sh) (u(xi +sh) - u,) ds = O(h)

-1/2

is attained for i = n, since u(:i: 71 +sh) = u(() + O(h) for any

s E [-0.5,0.5].

(2) If 8 > 0.5, then 1/J; = O(h^2 ) for if:- n + 1, while 1/J;+ 1 = 0(1) and

0

1/J~.+i = O(h).