1549301742-The_Theory_of_Difference_Schemes__Samarskii

Homogeneous difference sche1nes on non-equidistant grids 169

lead to the difference scheme

0 0

(2) - a-o z Yi -h Yi-1] - di Yi = - <pi ,

0

<pi

1

n 1

z

'"i+1/2

j J(x)dx,

'"i+1/2

J

(^0 1)
d; = Ji.
z
q(x)dx,
Xi-1/2
which is the best possible in the same sense as we spoke above about this
on an equidistant grid. On the same grounds as before, the coefficients ~ii
0 0
d; and <pi are representable by
(3)
0
a-z
0
d z
0
<p z
( 0 ) _,
f k(x;~shi)
-1
0
h·
J
1 h+1
q(x; + shi) ds + T
n
' -1/2 z
0
h·
J
(^1) J(x; + sh) hi+1

• n 1 ds + T
  z -1/2 '

1/2

J

q(x; + shi+i) ds,

0

l /2

J

f(xi + sh;+ 1 ) ds.

0

Retaining the notations of Chapter 2, Section 1

Yi - Yi-1

Yx,1: = h·

z

'1/ - Yi+1 - Yi

.x,i - h-

z+l

which allow a simpler writing of the ensuing formulae, we refer to the three-

point scheme

(4)
y(O)=u 1 , y(l)=u 2 , a > c 1 > 0 , d > 0.