1549301742-The_Theory_of_Difference_Schemes__Samarskii

160 Homogeneous Difference Schemes

is the error of approximation of equation (1) by the difference scheme (2)

on the solution u = 1t(x) of problem (1).

In Section 1.4 we have established conditions of the second-order lo-

cal approxirnation for the conservative scheme (2) with linear nonnegative

pattern functionals A[k(s)] and F[f(s)] such as:

(5) A[l]=l, A[s]=-!, F[l]=l, F[s]=O.

As can readily be observed, these conditions remain valid for any scheme

from the primary family.

In order to evaluate the order of accuracy for scheme (2), it is necessary

to make the accurate account of the error z = y - u being viewed as a solu-

tion of problem (3). Moreover, the desirable estimate should be expressed

in terrns of the right-hand side 1/;. In this direction the error of approxi-

mation to 1/;(x) is considered first. If k(x) E C(^3 ) and q(x), f(x) E cC^2 l,

then

1/;(x) =Au+ <p - (Lu+ f)

= [(aux) x - (ku')'] - (d - q) u + (<p- J) = O(h^2 ).

This means that scheme (2) provides a local approximation of order 2, so

that 111/J lie< Mh^2 , where M = const > 0 is independent of h.

In the sequel we succeed in showing that

111/J II. = ~~

1

h I k~l h 1/;k I < J'vi h

2

if k(x), q(x), f(x) E cC^2 l, meaning that k(x) possesses two (but not three)
continuous derivatives.

2. The error of approximation in the class of discontinuous coefficients.
   Our aim here is to justify that the error of approxirnation ( 4) can always
   be represented in the forn1
   (6) 1/; = 7/a: + 1/;*,

(7)

(8)

1/2

1/J7 = (<pi - j f(x; +sh) ds)

-1/2

1/2

( d; ui - j q(xi +sh) u(x; +sh) ds).

-1/2