1549301742-The_Theory_of_Difference_Schemes__Samarskii

192 Homogeneous Difference Schemes

By the same token,

so that

( 44)

1

h T' z

ri-1/2

1/J, * = 12r h2 ( qu - f)i I + 0( h2) ,

z

giving 11 r1/i* lie= O(h^2 ).

0

It is clear that the function 77; =a; ur,i - (ku');- 1 ; 2 = O(h^2 ), n1eaning

0

77i = ri-1/2 77; ,

0 2

(45) 77i = 0( h ).

Comparison of formulae (35) and ( 43) yields iJ = h, v. This provides sup-

port for the view that z; = O(h), because iJ = O(h^2 ) as stated above.

In order to estimate the enor z = y - u, we shall need an auxiliary

lemma.

Lemma Let z be a solution of problem (41) and let v be a solution of the
same problem for d; = 0, i = 1, 2,, .. , N - 1, q 0 = 0. Then the inequality
is valid:

(46) 11 z 11 c = 111ax I Z; I :::; 2 11 v 11 c ·
0'5oz<N
To prove this assertion, it suffices to use only the lemma of Chapter 1,
Section 1, Subsection 8 and to write down the equations for z and z - v in
the form ( 41).
The function vi can be recovered in explicit fonn from the conditions

Wi+1 - W;

----=-1fi,, i=l,2, ... ,N-1,

hr· z

that is,

where 1/ii is specified by formula ( 42) for di - 0. Summing the equations
wk+ 1 =wk - h rk 1/ik over k = 1, 2, ... , j yields

( 47)

j

W1+1 = w1 - L hrk1fk,

k=l

h2

W1 = -S ZJ.