1549301742-The_Theory_of_Difference_Schemes__Samarskii

486 Homogeneous Difference Sche1nes for Time-Dependent Equations

as required.

For the error z = y - u we may set up the related problem

One obvious way of proceeding is to introduce the space H of all grid

functions defined on wh and vanishing for i = N. Under the inner product

structure

N-l

[y, v) = L Yi V; h + 0.5 h Yo V 0

i=l

we refer to operators A and D acting in accordance with the rules

(Ay)i = -(Ay)i for 0 < i < N, (A y ) 0 = _al 0.5 Yx,o h ,

(D Y)i = Yi for^0 < i < N,
With the detailed forms in mind, problen1 (73) can be recast as
(74) Dz 1 +Az(a)=1/;, t=jr>O, z(O)=O,
showing the new members to be sensible ones. All this enables us too write
B = D + CJT A.
From the general operator theory outlined in Chapter 2, Section 4 it
seems clear that the operator D is self-adjoint and positive definite:
D>cE, where c=min(l,2C:/h).
However, this is certainly so with the operator A:
A= A> 0.
vVhat is more, the operators A and D are commuting: AD = DA. Due to
these properties the stability condition for scheme (74) is expressed by
B - o.5 r A= D+ +(CJ - o.5) r A> o,
which is valid only for
c
CJ>0.5- rllAll =Clo.
Ivlore a detailed proof of convergence of this scheme is concerned with
the form (74) and a priori estimates obtained in Chapter 6, Section 2 and
so it is omitted here. As a final result we deduce that scheme (70) converges
uniforn1ly with the rate O(r"'~ +h^2 ).