1549301742-The_Theory_of_Difference_Schemes__Samarskii

Higher-accuracy sche1nes 207

3.7 HIGHER-ACCURACY SCHEMES

1. An exact scheme. For equation ( 1) in Section 1 it is possible to make
   the design of a homogeneous conservative exact three-point scheme so that
   a solution Y; of the difference problem is identical with the exact solution
   1l = u( x) of problem ( 1) from Section 1 at all the nodes of any grid w h:

Yi = u(xi) for k,q,fEQ(o)[0,1].

Before giving further motivations, it will be convenient to set up problem

( 1) arising in Section 1 in the form

(1) L(p,q)u= !!_ (-

1

• du)-q(x)u=-f(x),
  dx p( x) clx
  O<x<l,

1

0 < p(x) < - ,

C1

u(O) = u 1 , u(l) = u 2 , p(x) = k-^1 (x), q(x) > 0.

It is worth noting here that the best scheme (14)-(15) of Section 2 is exact
for q = f - 0. Indeed, the function

(2)

x

u(x) = u 1 + c J p(i) di,

0

(

1 )-1

c= (u 2 -u 1 ) J p(i) di

0

is just the solution of problem (1) for q = f 0. From such reasoning it
seerns clear that

c

·u-x,'. = -h

0

a; u 53 i = C,

'

where ~i = (h-^1 I~ii-i p(i) di)-l and, therefore, function (2) solves the

equation (~u 53 ),,. = 0.
We now turn to equation (1) on an equidistant grid wh. A key idea in
the further development of an exact scheme is to express at any inner point
(and, particular, at x = x;) of the interval (xi-i, X;+ 1 ) a solution u = u(x)
to the second-order equation (1) in terms of the values u;_ 1 , ui+i and the
right-hand side f(x). This can be done using u(x) in the form

(3) u(x) =A; v; (x) + B; v~(x) + v;(x),