1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference schemes for the equation of vibrations of a string 367

Here the approximation error for the boundary conditions is a quantity

O(h^2 + r^2 ) if

<p=f(x,t),

The forthcoming substitutions

h2

O"=- 12 r-?+iT, iT =canst ,

h f31

P1 = 1 + -3-'

h^2 II

<p = f(x, t) + 12 f ,

h^2 .. (t)

V1(t) = μ1(t) + 6 ( μ^12 + f'(O, t) - f31 f(O, t))'

( ( h

2

V2 i) = μ2 i) + B ( μ2 ( i) / ( )

2

• f l, i) - f32 f(l, i)

are best suited for the design of the scheme of accuracy O(h^4 + r^2 ) approxi-

mating the initial equation at the nodes x = 0 and x = 1 to 0( ( h^4 + r^2 ) / h)

and the boundary conditions to O(h^4 + r^2 ).

2. Stability analysis. We now investigate the stability of scheme ( 4) with
   respect to initial data in the case of hon1ogeneous boundary conditions and
   zero right-hand side of the equation. A reasonable statement of the problem
   lS

( 4a)
Yo= YN = 0, y(x, 0) = u 0 (x),

On the same grounds as before, we look for its solution by the method
of separation of vahables, still using the framework of Section 1.4 where
particular solutions are sought as a product y(x,t) = X(x)T(t) "/:- 0. Sub-
stitution of y = X Tinto equation ( 4a) gives

(9)

A)<.

x

Putting these together with the boundary conditions y 0 = YN = 0 we set
up the eigenvalue problem for ,Y(x):

A,Y+.AX=O, X(O) = X(l) = 0,