1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternating direction inethocl 555

where

and

The value μ is put in correspondence with the expression

(35)

arising from the statement of Lhe difference problen1 (:33) during the course

of the elin1ination of A 1 y.

In this regard, it should be taken into account that sche1ne (33)-(34)

is equivalent to the factorized scheme (20)-(21) with the member Acxy =

(acx(x) Yx,Jx"' in special cases: either in scheme (33) the operator A2(f)

is involved at one and the same moment of time [in place of A2(tn) and

A2(tn+i) or kcx(x) and, hence, Acx are independent oft. Recall that scheme

(20) provides on a solution tt(x, t) an approximation of O(lhl^2 ) + r^2 if the

usual requirements of s1noothness of k er ( x) in the variables x 1 and x 2 are

imposed in addition to conditions (31). The principal difference from the

case of constant coefficients is discovered in the stability analysis of scheme

(20) by observing that the operators A 1 = -A 1 and A 2 = -A 2 are positive

and self-adjoint. But, unfortunately, they are non-com1nutative:

where ac, = acx(x, t). Just for this reason the product A1A2 is not obliged
to be positive, thus causing s01ne difficulties. In view of this, it is possible
to establish the stability only for sufficiently small values of T < r 0 (c 1 ),
where c 1 depends on the maximum of the derivatives of kcx with respect to
x 1 and x 2. Let us stress here that T < r 0 ( ci) is a severe restriction and, as a
matter of experience, it is connected with the available methods of special
investigations of stability. In what follows we will show that scheme (33) is
absolutely stable in another nonn.
With this aim, we first write clown the equation for the error by in-
troducing the new variables

• z=y-1t, - -