1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference schemes for the equation of vibrations of a string 369

By relating the initial conditions y^0 = u 0 and y~ = (y^1 - y^0 )/r = u 0 (x)

to hold for sum (11) we establish the relationships for determining the

coefficients Ak and Bk:

cos <pk - 1 B sm <pk

Ak T + k T

making it possible to find that

(13)

1 - COS 'Pk T _

Bk = llok + llok ·

sm 'Pk sm 'Pk

Having substituted Ak and Bk into (11), we are led by rninor changes to

. N-1( ~ cos (. J - 2 1) 'Pk
YJ = ~ i llok
k=l cos 2'Pk

+. T sin j<pk llok _ ) ){ (k)( x ).

Slll <pk

(14)

After the first stage the estimate of 11 yj 11 for scheme ( 4a) will be

derived for O" = 0 relating to the scheme

(15) Ytt = Ay, Yo=YN=O, y(x,O)=u 0 (x), Yt(x,0)=u 0 (x).

In that case

O:k -- 2 1 T 2 \ "k -- μk > cos 'Pk = 1 - μk '

When the steps of the grid whT are related by

(16)

T^2 1

• <
  h^2 -1+c:'

where E > 0 is an arbitrary number, we find that

2
( 17) μk < k=l,2, ... ,N-1,
~ 1 + E'

yielding

(18) cos 2 I 'Pk = vi r;-μ;l - 2 μk > v ~ 1+E.

By virtue of the relations

sm 'Pk

T

2 sin ~'Pk 1 > 2 sin ~'Pk Vrf

cos 2'Pk

T T l+E