1549301742-The_Theory_of_Difference_Schemes__Samarskii

374 Difference Schen1es with Constant Coefficients

In Chapter 2 it has been already shown that 4 h;^2 sin^2 ( ~ Irh 1 ) > 8 if

h 1 < ~. Therefore, keeping h 1 = 2h we find that

sin^2 7rh 4. 2 7rh 1

h^2 =-sm-> h^2 2 -^8

I

1

h < - 4 -.

Thus, if T <hand h < 1/4 the proper estimate for the solution of problem

(29)-(30) is

3. The energy inequality method. An investigation of difference schemes
   for the string vibration equation may be carried out by means of the energy
   inequality method (see Section 1). Here we restrict ourselves to stability
   with respect to the initial data with regard to the problem

(32)

Ytt = A (cry+ (1 - 20") y + O"Y) ,

Yo= YN = 0, y( X, Q) = u 0 ( X) ,

Bearing in n1ind that O" y + (1 - 2 O") y + O" y = y + O" r^2 Ytt and attempting

equation (32) in the form

(33) ( E - (} T^2 A) Ytt = A y

where Eis the identity operator, we take the inner product of the resulting
equation (33) and the quantity ya t = (Yt + Yr)/2. The outcome of this is

(34)

The trivial identities

help rearrange the left-hand side of equality (34) as

(35)

Further, we will show that for any function y = y( x, tn) vanishing at
the points x = 0 and x = 1