1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference schemes for the equation of vibrations of a string 365

A key role in subsequent discussions is played by a farnily of schemes with

weights

Ytt = A ( O" f; + ( 1 - 2 O") y + O" f;) + 'P ,

(4)

Yo = μ 1 (t), YN = μ 2 (t), y(x, 0) = u 0 (x), Yt(x, 0) = u 0 (x),

where u 0 ( x) will be specified below.

The boundary conditions and the first initial condition u( x, 0) = u 0 ( x)

on the grid whr are satisfied exactly. A choice of u 0 (x) is stipulated by the

wishes that the approximation error u(x) - 8u(x, 0)/8t = u(x) - uo(x)

would be a quantity of order O(r^2 ). From the chain of the relations

1

u 1 (x, 0) = u(x, 0) + 2" ru(x, 0) + O(r^2 )

= uo(x) + ~ T ( u"(x, 0) + f(x, 0)) + O(r^2 )

=u- 0 (x)+2T 1 ( uII )?

0 (x)+f(x,O) +O(r-)

it is readily seen that u( x) - ut ( x, 0) = 0( r^2 ) if we accept

(5) Uo(x) = iio(x) + ~ T (ui(x) + f(x, 0)) ·

Thus, the difference problem (4)-(5) is cornpletely posed. With regard to

ff= yj+i we set up on the basis of (4) the boundary-value problem

T

I= h'

O<i<N,

F; = ( 2 y/ - yf-I ) + T^2 (1 - 2 O") A yj + ()" r^2 A yj-I + r^2 'P'
which can be solved by the right elimination method, stable for any O" > 0
(see Chapter 1, Section 2.6).
The next step is to calculate the approximation error of ( 4) in the case
<p = f( x, t j) by investigating the difference between a solution y of problem
( 4 )-(5) and a solution cl = u( x, t) of problem (1 )-(3). Substituting y = z+u
into ( 4) yields

(6)
z(x,0)=0, z 1 (x, 0) = v(x),