1549301742-The_Theory_of_Difference_Schemes__Samarskii

396 Stability Theory of Difference Schemes

Scheme ( 4) provides an approximation on a function u( t) if

(31) max ll4Jj llc 2 l ---+ 0 as !hi----+ 0, r----+ 0.

o<;,1<;,10 h

Scheme (4) provides an approximation of O(lhlm + rk) on a function

u(t) E B~^1 l if

oT~x

0

111fijllc 2 h) < M(lhlm + rk),

_J _J

where M = const > 0 is independent of h and T both.

Along these lines, we 1nay set up the proble1n for the error zj = yj -u),:

z]+l_zj ..

B +Az^1 =·i/J^1 , j=O,l, ... ,

T

(32) z^0 =Yo - uh·^0

If scheme ( 4) is stable, then for a solution of problem (32) the estimate

holds:. Q ·/

llz^1 ll(h) < M1 llYo - uhll(h) + lvl2 0 9;;~j ll1fi^1 11(2h),

thereby justifying the next assertions.

Scheme (4) converges on a function u(t) ifit is stable, gen-

erates an approximation on u(t) and the initial value y 0

approximates the elen1ent u(O):

Scheme (4) is of accuracy O(lhlm + r") on a function u(t)

if it is stable, provides an approximation of O(lhlm + rk) on

u(t) and

In particular, u(t) may be a solution of a certain differential equation.
In that case we say that the difference scheme approximates the difference
equation, provided condition (31) holds, etc.
We note in passing that one is to understand the statement "if a
scheme is stable and provides an approximation, then it is convergent"
given in Chapter 2, Section 2 as follows: both the difference equation and

the initial value generate an approximation (if we accept y 0 = P~^1 )u(O),
then II YohT - uh(O) ll(lh) = 0).