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Volume 1, 1997 |
18 |
[4] T.S. Vashakmadze, A Generalized Finite-Difference Method, Differential Equation, vol. II, N 5, 1966, 614-618 pp.
ON A STABILITY OF MULTI-STEP METHOD FOR AN
ABSTRACT PARABOLIC EQUATION
J. Rogava
Georgian Technical University
In the Hilbert space H for the Cauchy problem
(1) u'(t)+Au(t)=f(t), t>0, u(0)=j ,
where A is self-adjoint, positive definite operator, the multi-step method defined by means of the following scheme is considered:
| (2) | |
where q ³ 1, 
>0; uk is the approximate solution of the problem (1) in t=tk=k; u0, u1,..., uq-1 are given initial vectors from H; a i and b i (i=0,1,...,q) are real numbers, and a q >0, b q >0.
In order to study the stability problem for the scheme (2) we introduce a special class at polinomials, which is defined by the following recurent dependence:
Uk=x1Uk-1+x2Uk-2+...+xqUk-q,
U0=1, U-1=...=U1-q=0.
On the basis of the properties of these polinomials the following theorems are proved.
Theorem 1. If the multi-step method (2) is stable, then all the roots of the characteristic equation
(3 ) l 2-x1(s) l q-1-...-xq-1(s) l -xq(s)=0, sÎ [0, +¥ )
where xi(s)=( a q-1+b q-1 s)( a q +b q s)-1, i=1 ...,q, belong to the unit circle.
Theorem 2. Let for arbitrary sÎ [0, +¥ ), all the roots of the equation (3) but probably one, belong to the same circle which is inside the unit circle. Then the multi-step method (2) is stable.