1549301742-The_Theory_of_Difference_Schemes__Samarskii

416 Stability Theory of Difference Schemes Stability of the weighted scheme. As an example of applying the theorems just establ ...

Classes of stable two-layer schemes 417 Theorem 8 Let A be a self-adjoint positive operator independent oft = nr; A = A* > 0. ...

418 Stability Theory of Difference Schernes The condition a-> a- 0 implies that (52) B--A>sA-. - T -^1 2 - Indeed, for a- ...

Classes of stable two-layer schemes Indeed, putting x = A-^1 y we obtain from (55) the inequality (y, y) < 6.(A-1y, y), meani ...

420 Stability Theory of Difference Schemes Joint use of Lemma 6 and inequality (60) permits us to state the following. Theorem 1 ...

Classes of stable two-layer schemes 421 As before we assume the existence of an inverse operator B-^1 (t), which assures us ofso ...

422 Stability Theory of Difference Schemes provided condition (64) holds. The energy identity for t = 0 admits the form 2r ((B(O ...

Classes of stable two-layer schernes 423 Example. In order to apply the general stability theory for particular difference sche ...

424 Stability Theory of Difference Schemes After that, taking into account that Yi-1 = -h Yx ,i +Yi , Yi+1 = h Yx ,i +Yi , h Yx ...

Classes of stable two-layer schemes 425 Together with scheme (71) one can write down one more asymmetric scheme which after redu ...

426 Stability Theory of Difference Schemes that is, there exists an inverse operator B-^1. By inserting y = B-^1 x it is plain t ...

Classes of stable two-layer schem.es 427 This implies the estimate for a solution of the nonhomogeneous equation Byt + Ay = < ...

428 Stability Theory of Difference Schemes 6.3 CLASSES OF STABLE THREE-LAYER SCHEMES The problem statement. In this section we ...

Classes of stable three-layer schemes 429 (A, B and Rare, in general, variables, that is, they depend on tn)· From such reasonin ...

430 Stability Theory of Difference Schemes of Yo, y 1 , ip( t) such that for any Yo, Y1> ip( t) and all t = T, 2r, ... , ( n ...

Classes of stable three-layer schemes 431 we rep resent scheme ( 1) in the form ( 11) y(O) = Yo , Y( r) = Y1 , where A= A(tn) =A ...

432 Stability Theory of Difference Schemes By inserting in (16) v = fJ and z = y we transform (15) into (17) (A(fJ + y), fJ - y) ...

Classes of stable three--layer schemes 433 In our basic account A and R are taken to be constant self-adjoint positive operators ...

434 Stability Theory of Difference Schemes where II Y II is defined in accordance with rule (24). Indeed, for B > 0 identity ...

Classes of stable three-layer schemes is equivalent to the require1nents (26*) (27*) B = B(t) > 0 for all t E W 7 , 1 R>-A ...