1549301742-The_Theory_of_Difference_Schemes__Samarskii

444 Stability Theory of Difference Schemes

while the estimate of the form (61) is valid for

7. Examples. It seems worthwhile giving several schemes of particular
   forms.
   
   
   1. Scheme ( 1) with the operators R = u E and B = E
   
   
   
   

(79) Yt + u T^2 Ytt + A y = 'P

is stable for uE > A/4, that is, for

(80) u > i llAll.

A particular case of scheme (79) is the Du Fort-Frankel scheme known
as the "rhon1bus" scheme for the heat conduction equation

OU

at

cPu

8x^2 '

0 < x < 1, t > 0, u(x, 0) = u 0 (x),

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

emerging from the explicit unstable scheme

Yr+ Ay = 0'

y(x + h, t) - 2y(x, t) + y(x - h, t)

Ay = -Yxx = h2 '

upon replacing y(x, t) = y/, by the half-sum ~ (yf +^1 + yf_-^1 ) = ~ (fJ; + fJ;),
The outcome of this is

( 81)

Y;+i - (f;; + Y;) + Yi-1

h2

Also, it will be sensible to write (81) in canonical form. Since

y ~ + y ~ =^2 y + T^2 Ytt )

T2
the right-hand side of (81) equals Yxx - h 2 Ytt· Therefore,

T2

Yr+ h 2 Ytt + Ay = 0, Ay = -Yxx,

4 27rh 4

11 A 11 = h 2 cos 2 < h 2.