1549301742-The_Theory_of_Difference_Schemes__Samarskii

Methods for designing difference schen:1es 223

In light of the properties of the function 7Ji ( x) and its derivatives the matrix

{ CYij} is tridiagonal, because only the ele1nents with J = i - 1, J = i and

J = i + 1 are nonzero.

In the new notations

(29) ai = -hCYi i-1,

'

we obtain

Xi x· z

1

J

1

J

a·= - k(x)dx--

' h h

q(x) (x - X;_ 1 ) (x; - x) dx,

'"i-1 Xi-1

(30)

q(x)(x - x;_,) dx + 7' q(x)(x;., - x) dx).

x;

Then the system of equations

CY; ' i-1 Yi-1 +CY; ' i Yi+ CY; ' i+1 Yi+l - f3; = 0

can be rewritten as

or

(31)

where

( J

Xi-I

"'i+I

f(x) (x - xi_ 1 ) dx + j f(x) (x;+ 1 - x) dx) ,

Xj

(32)

thereby clarifying that tp; are calculated by means of the same formula as
d;.
So, the three-point scheme (30)-(32) constructed by the Ritz method
is identical with scheme ( 12) obtained by means of the IIM. In contrast
to the Ritz method the Bubnov-Galerkin method applies equally well to