1549301742-The_Theory_of_Difference_Schemes__Samarskii

396 Stability Theory of Difference Schemes Scheme ( 4) provides an approximation on a function u( t) if (31) max ll4Jj llc 2 l - ...

Classes of stable two-layer schemes 397 6.2 CLASSES OF STABLE TWO-LAYER SCHEMES The problem statement. We pursue the stability ...

398 Stability Theory of Difference Schemes When the triangle inequality II Y ll(ll <II 'fJ ll(ll +II fJ ll(ll is put together ...

Classes of stable two-layer sche1nes 399 err A. Since A< 11A11 E and E >A/II A 11, we have B > (1/11 A ll+err)A > 0 ...

400 Stability Theory of Difference Schemes Theorem 1 The condition ( 14) B >!..A - 2 is necessary and sufficient for the stab ...

Classes of stable two-layer schemes^401 Remark 1 Condition (14) is sufficient for stability in the sense of (15) of scheme (1) i ...

402 Stability Theory of Difference Schemes With the relations A< II A llE and E >A/II A II in view, we obtain B-0.5rA> ...

Classes of stable two-layer schemes 403 The operator A - = E = A* - > 0 is independent of t and the operator B - is positive ...

404 Stability Theory of Difference Sche1nes and relation (20) yields ll:iills < llYll 8 , meaning II y(t) lls < II y(O) ll ...

Classes of stable two-layer schernes 405 The outcome of further substitutions x = B-^1 Ay or y = A-^1 Bx is the same: (23) J A [ ...

406 Stability Theory of Difference Schemes Thus, the functional J B [y] is transformed into the same form (23) as was done befor ...

Classes of stable two-layer schemes^407 we find upon substituting (28) into (la) that Putting this together with the orthogonali ...

408 Stability Theory of Difference Schemes Let us show that inequalities (14) and (30) are equivalent. To this end, we operate w ...

Classes of stable two-layer schemes^409 provided that the forthcoming substitutions are carried out before going further: 1) Xn ...

410 Stability Theory of Difference Schemes thereby justifying coinciding of the signs of the opera.tors C - / E a.nd E - ~fc-^1. ...

Classes of stable two-layer schemes 411 are necessary and sufficient for the p-stability in the space H B of scheme (la): 11 Yn ...

412 Stability Theory of Difference Schemes If, in addition, the operator B is self-adjoint, then n II Yn+I lls < II Yo lls + ...

Classes of stable two-layer schen1es 413 with C = A^1!^2 B-^1 A^1!^2 enables us to evaluate the smnmand in inequality (41). Stab ...

414 Stability Theory of Difference Schemes Proof The energy identity (13) is involved at the first stage. Estimation of its righ ...

Classes of stable two-layer schemes 415 Furthermore, we substitute this estimate into ( 13). Having stipulated con- dition (44), ...