1549301742-The_Theory_of_Difference_Schemes__Samarskii

The transfer equation 363

holds true due to the relation YYxx = ~(y^2 )x + ~h(y:f:)^2. An alternative

form of scheme (12) rnay be useful in the further development:

(16) ( E + O" T A) Yt + A y = 0.

Since A> 0, there always exists the inverse Jr-^1 > 0, by rneans of which

we recast (16) as

(17) ( A -l + O" T E ) Yt + y =^0.

Comparison of (17) with (14) shows that B = J-^1 + O"T E, A

con di ti on ( 15) signifies that

((J-^1 + O"T E) x, x)-~ r(x, x)

E and

(

= A--^1 x, x) +(O"-^1

2

) T (x, x) > 0,

yielding (A y, y) + ( O" - ~) T 11 A y 112 > 0 for the substitution J-^1 x = y and
providing the relation ( Ay, y) = ~ y'jy + ~ h II Ay 112 in the case of interest.
The stability condition (15) taking now the form

(18)

is valid for O" > 0" 0. This provides enough reason to conclude that the a
priori estimate II yj II < II y^0 II is certainly true. Likewise, a scheme with
weights may be designed for the systen1

au

. 75i

av

ox '

av

at

OU

ox '

which is equivalent to the equation of vibrations of a string

82 ti 82 u

8t^2 8x^2

and the stability conditions for such a scheme rnay be established rn a
similar way as we did before.