1549301742-The_Theory_of_Difference_Schemes__Samarskii

216 Homogeneous Difference Schemes

and taking into account (3), we establish

(5)

1

2h

X;'i+l

(qu - .f) dx.

The approximation of the integral on the right-hand side of (5) can

be done using various quadrature formulae, for instance, by the formula of

trapezoids

or

1

2h

Xi+!

2

1

h j (qu - .f) dx i':::! (qu - f)i

x· i-1

Xji+l 1 •

(qu - .f) dx i':::! 4 ((qu - .f)i-1 + 2 (qu - f); + (q·u - f)i+i)

h2

= (qu. - f)i + 4 (qu - f);;;x,i.

With these, we arrive at the schernes of accuracy O(h^2 ):

(6) (a Yr:)·'" - q y = - .f ,

(7)

h2 h2

( ayx - 4 (q y)x) x - qy = -(.f + 4 fxx).

To approximate the boundary condition, for instance, at the point

x = 0, we apply (3) for i = 0

h

w 1 - w 0 = J ( qu - J) dx

0

and then adopt here

h

w 1 i':::! a 1 ux,o+ ~ J(qu-f) dx,

0

so that
h
a1 ux,O - (u111.o - μ1) i':::! ~ J (qu - .f) dx.
0