1549301742-The_Theory_of_Difference_Schemes__Samarskii

The transfer equation 359

The residual on a solution u(x, t) is

Q.5 h^2 1 .. 1 h^2 II ( 2 2)

1/J = Ut + au x 0 - T U;:x. = - 2 Tu - - 2 - T u + 0 h + T.

With the members recovered from the equation ii = a^2 1l^11 , the preceding

becomes

1/J= (a~T - ~;) u"+O(h^2 +T^2 ),

which provides support for the view that scheme (8) generates a conditional

approximation, because it approximates the equation only if h^2 jT--+ 0 as

h --+ 0 and T --+ 0. In my view, two important things are that for T = 0( h)

we rnight achieve 1jJ = 0( T + h) and the second order of approximation

·ij; = O(h^2 ) is attained for T = h/I a I·

To make our exposition more complete, we involve the four-point sche-

me of second-order accuracy

(9) Yt +a Yg, - 2 I Ta^2 Yx:c = Q

on the same pattern as was done for scheme (7) (see Fig. 17 c). Plain

calculations of the residual with u + au' = 0 and ii = -au' = a^2 u"

incorporated give

of, '// = ut + a Uo - - 1 2

2

x Ta 'tlxx

= ( il + Cl u') + ~ T (ii - a^2 '1l^11 ) + 0( h^2 + T^2 ) = 0( h^2 + T^2 ) '

so that ij; = O(h^2 + T^2 )

Let us investigate the stability of scheme (9) by the spectral rnethod

having represented it beforehand in the form

Yk j+1 = ( 1 - I 2) Yk j + 2 1 ( I I - 1 ) Yk+1 j + 2 1 ( I I+ 1 ) Yk-1 j ·

Via transform Yf = q j eiktp we derive the useful expression for q:

q = 1 - 12 ( 1 - cos <p) - i I sin <p , I q 12 = 1 - I 2( 1 - 1 2)( 1 - cos <p )2 ,

by means of which we establish that I I I < 1 is a necessary stability condi-
tion, whereas Iii > 1 is a sufficient unstability condition in the situations
when Iii is fixed and Iii = canst with varying parameters T and h.