1549301742-The_Theory_of_Difference_Schemes__Samarskii

Convergence and accuracy of homogeneous conservative schemes 165

where 1/;(x) is the error of approximation having the form (6): 1/; = 7/x + 1/;*.

We will show that the solution of problem (3) with the right-hand side (6)

admits the estimate

(12)

where μi = L~~\ h 1/JZ for i = 2, 3, ... , N and μ 1 = 0. To that end, it

suffices to carry out the proper evaluation of a solution of the auxiliary

problem

(13) v(O) = v(l) = 0.

Other ideas are connected with solving the difference problem

w(O) = w(l) = 0,

frorn which follows i mn1ediately that awx + 17 = con st = c 0. As its corollary,

we establish the recurrence relation for wk:

Summation over k from 1 to i yields

i

W; = Wa - L

k=l

k=l,2, ... ,N.

Putting i = N and keeping in mind that w 0 = wN = 0, we find that

yielding

or

'W; = - 2=

k=l