1549301742-The_Theory_of_Difference_Schemes__Samarskii

156 Homogeneous Difference Schemes

In practice it is convenient to employ more simpler formulae for de-

termination of a, d and <p with using the values of k, q and f at the isolated

points. Usually a pattern consisting of one or two points permits one to

make considerable simplifications:

(19)

or

ai = ki_ 112 = k(xi - 0.5 h)

'Pi = Ji

(A[k(s)] = k(-0.5)),

(F[f(s)] = f(O))

(A[k(s)] = ~ (f(-1) + k(O)),

Conditions (18) are certainly true for all the schemes we have mentioned

above.

If the coefficient k( x) is discontinuous at the middle nodal points x =

xi_ 112 and the coefficients q( x) and f ( x) are discontinuous at the points x =

xi, the half-surns of the left and right limiting values have to be substituted

into formulae ( 19). As a final result we get

ai = ~ (k(x; 112 - 0) + k(xi 112 + 0)),

di = ~ (q(xi - 0) + q(x; + 0)),

'Pi= ~ (f(x; - 0) + f(x; + 0)).

We note in passing that formulae (19) and some others for determination
of the coefficients a, d and <p can be derived through the approximations to
integrals (15) and other members

Xi

~ J

dx 1

--~ ---

k(x) ki-1/2 '

:L'i-1

4. A primary family of conservative schemes. We spoke above about the
   family of the homogeneous conservative schemes (17), whose description is
   connected with some class of pattern functionals A [k(s)] and F [f(s)]. For