Economical factorized schemes

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Knowing the value y = on the 7th layer, it is required to find the value 2/- +. In preparation for this, we derive the equation related to y with a known right-hand side  

As can readily be observed, 0 N) operations are needed in giving F and their amount is proportional to the total number of the grid nodes. This is certainly so with any difference scheme, whose pattern is independent of the grid. From equation (2) it is easily seen that the stable scheme (2) will be economical once the users perform 0 N) operations while solving equation (2). 

Let economical operators Ba, a = 1,2. p, be. such that 0(N) operations are necessary in connection with solving the equation 

Then scheme (1) with a factorized operator B of the structure 

Especial attention is being paid to the factorized schemes with the members 

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Constructions of economical factorized schemes. Using the regularization method behind, we try to develop the general method for constructing stable economical difference schemes on the basis of the primary stable scheme... 

Under such an approach the factorized scheme is of no less than first-order accuracy in r. In a similar way an economical factorized scheme can be designed in the p-dimensional case when... 

A particular case where R = a-A, cr = 0.5 (cr +cr ), is showing the gateway to the future research, whose aims and scope are connected with the general method for constructing three-layer economical factorized schemes by means of the regularization principle of difference schemes. A simple example... 

The algorithm of solving equation (45) was demonstrated before and so it remains only to construct economical factorized schemes associated with problem (42) by means of the operator acting in accordance with the rule... 

The problem statement. First of all, it should be noted that it is impossible to generalize directly the alternating direction method for three and more measurements as well as for parabolic equations of general form. Second, economical factorized schemes which have been under consideration in Section 2 of the present chapter are quite applicable under the assumption that the argument x = (xq, x, ., Xp) varies within a parallelepiped. 

If G is a parallelepiped in the space RP, problem (63)-(65) can be solved through the use of an economical factorized scheme with accuracy 0(r + /i ). The design of such a scheme was made in Section 2 and it was investigated there in full details. Applying the same procedure serves ti motivate that, first, the operators... 

When Qp u(, . = 0(1) is accepted in some suitable grid norm (2 ) built into stability theorems, we might achieve second-order accuracy in r as stated before due to the extra smoothness of the solution u. Such a stability analysis of schemes (36) and (39) is mostly based on the further treatment of the operators R and A as linear operators acting from... 

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