1549301742-The_Theory_of_Difference_Schemes__Samarskii

Heat conduction equation with constant coefficients

one succeeds in showing that J > 0 for(}> () 0. Because of (45),

l= llvll^2 +((}-(}0)^7 llv:cJl^2 +((}0- ~)r llvz]l^2

>II vll^2 - ~ h^2 II vz]l^2 > 0

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for (} > () 0 = ~ - h^2 ( 4r )-^1. Co1nbination of the preceding relations reveals

a profound result known as the energy inequality:

If zp = 0 and y is a solution of problem (16a), then 11 y~+^1 ]I< ···<II y~]I,

meaning that for (} > () 0 scheme (16) is stable with respect to the initial

0

data in the norm 11 y 11(1) = 11 Y.7:] I, which is a grid analog of the W }-norm.

No doubt, it sounds interesting, but the discovery of stability with respect

to the right-hand is of special merit in this matter. Letting

1

( 46) -

2

(l-c:)h^2

4r

we claim that

( 47)

Indeed,

for

J = 11 v 112 + ( (} - (Jc) T 11 Vz l 12 + ((Jc - ~ ) T 11VzJ1^2

llvll^2 - (l-:)h

2

llvxll^2

II v 112 - (1-c:) II v 112 =c:IIv11^2 ·

Substituting ( 47) into ( 44) yields the energy inequality

( 48)

By applying successively the Cauchy-Bunyakovski'l inequality and the c:-
inequality we arrive at the chain of the relations