1549301742-The_Theory_of_Difference_Schemes__Samarskii

636 Econmnical Difference Sche111es for M ulticli1nensional Proble111s Additive schemes for a syste111 of equations. Later in t ...

The summarized approxiination method 637 reduces to successive solution of the syste1n of 2p equations (85 ) l 2p av 8t = L-er v ...

638 Economical Difference Schemes for Multidimensional Problems The initial condition is satisfied exactly: (88) y ( x , 0) = u ...

The sun11narized approxiination 1nethocl 639 In concluding this chapter it will be sensible to introduce the space 0 rt of all g ...

640 Econorn.ical Difference Schernes for Multidi1nensional Proble1ns It follows from the foregoing that 0 (A-y,y) = (A+y,y) = 0. ...

The summarized approximation inethod 641 We will pursue the discussion of additive schemes further with re- gard to problen1 ( 6 ...

642 Econmnical Difference Schemes for Multidimensional Problems vVith regard to yj+i = Y(p) we obtain the syste1n of equations ( ...

10 Methods for Solving Grid Equations In this chapter econo1nical direct and iterative methods are designed for nmnerical soluti ...

644 Methods for Solving Grid Equations example, for any grid with steps h along all the directions x 1 ,x 2 , ... ,xP (h 1 = h 2 ...

Direct methods^645 The statement of the difference Dirichlet problem associated with problem ( 1) is (2) Ay= -f(x), y/,h =p(x), ...

646 Methods for Solving Grid Equations Also, it will be sensible to deal with the newly formed vectors j=l,2, ... ,N2-l, Fj = (P ...

Direct n1ethods^647 Having recovered from the systern (6) the vectors Yi with even nutnbers, we succeed in finding these with od ...

648 Methods for Solving Grid Equations or, what amounts to the same, 2v(ll - h 2 2 v~.r1:r1^1 ) - r 'ti vCll = v(l-l) ' h 1 < ...

Direct 1nethocls^649 showing the new vectors to be sensible ones. Summarizing, the numerical solution of problem (2) can be done ...

650 Methods for Solving Grid Equations The inet.hod of separation of variables. The problen1 we must solve is problem (2) with ...

Direct 111ethocls^651 where 'Pk ( ih 1 ) is the Fourier coefficient of the function 'P( x): N2-1 'Pk(ih1) = ~ 'PCih1,Jh2)μk(jh2) ...

652 Methods for Solving Grid Equations where Yj and Fj are vectors of the same order Mj, Cj is a square matrix of size Mj x M 1 ...

Direct methods 653 Here Yj and f3j are vectors of order N 1 - 1, Ci'j and Care square matrices of the same size ( N 1 - 1) x (N ...

654 Methods for Solving Grid Equations Before going further, we recall that any linear one-step iterative r11e- thod can be writ ...

Two-layer iteration sche111es 655 The usual practice involves the numerical solution of problem (5) with a prescribed accuracy c ...