1549301742-The_Theory_of_Difference_Schemes__Samarskii

164 Homogeneous Difference Schemes

It follows from the foregoing that

(10)

17; = O(h^2 )

1/J; = O(h^2 )

0

1/;~ = O(h)

1/J; = O(h^2 )

0

for i #- n + 1, 17n+ 1 = 0(1),

for ii- n, 1/;~ = 0(1),

if 8<0.5,

for ii- n + 1, 1/;~+ 1 = 0(1),

1/;~+ 1 = O(h) if e > 0.5.

0

17,,+ 1 = O(h),

Using these estimates behind we draw the conclusion that at the nodes

x = x,, and x = x,,+ 1 the function 1/;( x) can be expressed by

(11) 1/;,, =^17 nh+l + 0(1), 17n+l = 0(1)'

thereby clarifying that at the nodes adjacent to the discontinuity point
x = ( scheme (2) does not approximate equation (1) in light of the limit
relations

as h--+ 0. Fron1 asyn1ptotic formulae (11) it is readily seen that the main
summands in the expressions for 1/;,, and VJn+l have equal rnodules and
opposite signs, so that

meaning that the error of approximation is of di pole character in neighbor-
hoods of discontinuity points of the coefficient k(x). This provides enough
reason to conclude' that the conservative scheme (2) is of order 1 in the
norm

111/Jll. = (1, 1771] + (1, lμIJ = O(h)

with μi = L~~\ h1f;~ for i = 2, 3, ... , N and μ 1 = 0.

3. A priori estimates of the error. We now estimate the error z = y - u,
   which is a solution of problem (3):

Az = (az,.)x - dz= -1/;(x), 0 < x = ih < 1, z(O) = z(l) = 0,