1549301742-The_Theory_of_Difference_Schemes__Samarskii

Heat conduction equation with constant coefficients 311

So, not only the order of approximation, but also the stability of

scheme (16) depend on the pararneter (}.

While investigating stability we dealt actually only with two time

layers t j, t j +i and the step T = t j +i - t j. All the tricks and turns remain

unchanged when the grid w 7 becomes non-equidistant, that is, the step

TJ+i = tj+i - tj depends on the number of the layer. In this situation the

parameter(} may depend on the number j + 1 of the layer, say(}= (JJ+^1 ,

which serves to motivate the presence of another condition (} > (Jt+^1 =

~ - h^2 /(4rj+i) in place of (23). For instance, for the scheme of accuracy

O(h^4 + rf+ 1 ) we may accept (JJ+^1 = ~ - h^2 /(12rj+ 1 ). The condition

(} > (Jt+^1 is sufficient for the stability of the scheme with weights in the

case when the grid w 7 is non-equidistant.

5. Stability with respect to the right-hand side. As a rnatter of fact, if
   (} > 0, condition (23) given by

1 h^2

4T '

is sufficient for the stability of scheme (16) with respect to the right-hand

side as well. It is interesting to consider problem (16b) and look for its

solution in the form

N-l

(26) fJ = 2= tk x (k),

k=l

by representing the right-hand side <pin terms of the eigenfunctions {X (k)}:

N-l N-l

(27) <p= L'PkX(k), II 'Pll^2 = 2= 'P~.

k=l k=l

Substituting (26) and (27) into (16b) and recalling that A,y(k) = -,\kX(k),

we derive the series

N-1

L { Tkt (l +(}TAJ+ Ak Tk - 'Pk } x (k) = 0,

k=!

from which it follows that the expression in the curly brackets vanishes as
far as an orthogonal system of eigenfunctions is concerned:

(28)

l-(l-(J)TAk

1 + (JTAk