1549301742-The_Theory_of_Difference_Schemes__Samarskii

Heat conduction equation with constant coefficients 303

The six-point symmetric scheme with (} = 0.5 ascribed to Crank and

Nicolson is of the form

y/+1 - y/

T

(10)

3. The error of approximation. In order to evaluate the accuracy of scheme
   (4)-(6), the solution y = y/ of problem (4)-(6) should be compared with
   the solution u = u(x,t) of problem (I). Since u = u(x,t) is the continuous
   solution of problem (I), we may set u/ = u(x;, tj) and deal then with the
   difference z/ = y/ -u/. For this, the first step in the estimation of the grid
   function z/ on the relevant layer is connected with norms II · II of proper
   form, for example,

llzll=llzllc= O<i<N max lz;I, (

N-1 )1/2

llzll=

2

~ z?h

Let us pass, time and again, to the notations without subscripts and su-

perscripts, which are good enough for our purposes:

yj z = y, '1/j • z +i = YA , Yt = ( fJ - y) IT ,

permitting us to recast the problem we have completely posed by conditions

(4)-(6) as

Yt = A ( (} fJ + ( 1 - (}) y) + 'P ,

(II) y(O, t) = u 1 (t), y(l, t) = tl 2 (t),

Y( X, 0) = U 0 ( x) ,

(x, t) E whr,

Ay = Yxx ·

In trying to establish the conditions for determination of z = y - u we
substitute y = z + u into (II) and regard u as a known function, making it
possible to set up the problem for z:

(III)

where

( 11)

zt =A ( (} z + (1 - (}) z) + 1/J,

z(O, t) = z(l, t) = 0,

z(x,0)=0,

(x,t)Ewhr>