1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable three-layer schern.es 451

Applying now the inequality ab < 8a^2 + 62 /( 48) yields

(94)

Assuming the second coefficient to vanish, we deduce that /j = 1/(1 + c:)

and

J > - (^1) +c: E II y II~.
The second estimate of the lemrna is proved.
In order to obtain the third estimate, we require the equality of the
coefficients at the members II f; II~ and llYill~ in (94); this yields
1
/j - ---r===

• ,;1 +c:,

l _ /j = Jl + E - 1 = E

Jl + E 1 + E + Jl + E

Since Jl + c: < 1 + c: for any c: > 0, it follows from the foregoing that

1 - fJ > 2 ( 1 °+<) and

Thus, the lemma is completely proved.

Upon substituting (91)-(93) into (88) we obtain estimates for problem

(84a)

(95) llYn+rllA(tn) <Mr P (1/Yol/A( 7 ) + llY1(0)11n( 7 ))'

(96) llYn+r//A(t,.) + llY1lln(tn) < 2Mr p (11Yol/A( 7 ) + llY1(0)//n( 7 )) ·

In order to prove the stability of scheme (84) with respect to the right-
hand side, let us employ the superposition principle and seek a solution of
problem (84b) as a sum

n

(97) Yn = I: T gn s 1

s=I '

n= 1,2, ... , Yo= 0, s = 1, 2, ... '

where gn s as a function of n for any fixed s satisfies equation (84a) and
the initial ' conditions

(o.5 T B(t.) + D(ts)) gs > s = 0 ·