1549301742-The_Theory_of_Difference_Schemes__Samarskii

696 Methods for Solving Grid Equations The constants f; and ~ remain as yet unknown. Since f; 1s the smallest eigenvalue, it is ...

The alternative-triangular inethod 697 The calculations start from the corner i 1 = l, i 2 = 1, ,. , , iP = l, of the parallelep ...

698 Methods for Solving Grid Equations with (65) Asm cx(3^1 j m = 2 ~ [(ksm cx(3 Yx~ m) :ra + (ksm cr(J Yx~ m) .r 0 ] ' making i ...

The alternative-triangular method 699 where f;ij is, as usual, Kronecker's delta. We learn from Chapter 9, Section 2 that the co ...

700 Methods for Solving Grid Equations Current explanations need certain clarification by having recourse to one simplest exampl ...

The alternative-triangular 111ethod 701 conditionality of the systen1 (72) of equations along with decreasing the ratio h 1 / h. ...

702 Methods for Solving Grid Equations by means of which the factorized operator B is selected by the rule (76) where w > 0 i ...

The alternative-triangular method 703 lines x°' = x~"), i°' = 0,±1,±2, ... , Ct= 1,2, the points xi= (x~ii),x~i^2 )) constitute ...

704 Methods for Solving Grid Equations where a+ Q =Cl Ct' (xC+lac)) l l Yx" = h+ (y(+lac) -y)' °' = _l (y'(+la) _ y). Y.i" fi°' ...

The alternative-triangular 1nethod^705 Cumbersome calculations for a proper choice of d(x) will be excluded from further con.sid ...

706 Methods for Solving Grid Equations k+l For determination of y we must solve the equation k+l k (D+wA1)D-^1 (D+wA2) y = F k k ...

The alternative-triangular method 707 where fia = h and h! differ from h only at the near-boundary nodes, the intention is to us ...

708 Methods for Solving Grid Equations This provides enough reason to conclude that in the case of any com- plex d0111ain the nu ...

The alternative-triangular method 709 But it may happen that the coefficients of the governing differential equa- tion are varyi ...

710 Methods for Solving Grid Equations Adopting those ideas, the frainework of ATM (see Section 7) involves the operator B = (E ...

Iterative alternating direction rnethocls 711 The numerical solution of problem (1) by means of iteration schemes can be done us ...

712 Methods for Solving Grid Equations homogeneous equations for the error zi +^1 = yi +^1 - u: zi+l/2 _ zi (1) Tj+l zi+l _ z)+l ...

Iterative alternating direction rnethods 713 Also, we take for granted that (5) 1) (6) 2) the operators Ai and A2 are commuting ...

714 Methods for Solving Grid Equations since the operators Ai and A2 are commuting. As a final result we get n (8) Zn = Tn Zo , ...

Iterative alternating direction inethods^715 The norm II Tn II can be expressed through the eigenvalues of the operators A~ and ...