1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference schemes for the equation of vibrations of a string 373

By exactly the same reasoning as before we deduce for problem (4) with

zero boundary conditions y 0 = YN = 0 that

11111 II < ~ (II y' II+ 1111i 11,-' + j~ T 11~^1 ' II,-·)

under condition (21) and the restriction O" > 0. Intuition suggests and in

this case it does not deceive us that for a special choice y^1 = y( r) the

stability of scheme (15) under the constraint T < h may be proved in the

space L2.

We learn from Section 5.1 that the difference scheme

(29) Ytt j -- yj 'fa , j = 1, 2, ... ,

(30) Y^0 - - Uo , Yt^0 - - Uo - + 2 l T Yxx^0 ·

provides an approximation of O(h^2 + r^2 ) to equation (1 )-(2). Other ideas

are connected with expressions of the solution y j in terms of y^0 = u 0 and

il 0. Following established practice, we find that

(31) y^1 = 1t

1

(uok cos j'Pk + '.ilok sin j'Pk) x(k),

k=l sm 'Pk

where il 0 k are Fourier coefficients of ft 0 (x), the sense of the quantities u 0 k

and 'Pk being reserved. Squaring (31) and applying the estimate

(^2) ( T Uok cos j'Pk ) u 0 k sin j'Pk < T^2 u;k cos^2 J'Pk · + ·u 02 k sm ·^2 J'Pk · ,
sm 'Pk sin^2 'Pk
we obtain
N-1
11 Y j 112 < 11 Uo 112 + T^2 L
k=l
To evaluate the lower bound of the expression
1
sm. 2 'Pk - /\ \ k (1 - 4 l T 2 \ ) , /\ k
we take I = T / h <" l. Along these lines, it is evident that
Ak ( 1 - T24.>,k ) =
4
h2 sin^2 --Irkh ( 1-I 2. 2 sm --Irkh)
2 2
4

h2 sin^2 --Jrkh ( 1-sm.^2 --Irkh)
2 2
4
sin^2
Irkh 2 Irkh
-- cos --
h2 2 2
4
sin^2
Irh
cos^2
Jrh sin^2 Irh
h2
2 2 h2