Difference scheme economical

First of all observe that if scheme (108) or (109) with an operator R is stable, then so is a scheme with an operator R > R Common practice in designing difference schemes involves the development of a scheme which generates an approximation of the attainable order and is economical. With the indicated properties, its stability will be given special investigation. 

Economical Difference Schemes for Multidimensional Problems in Mathematical Physics... 

To understand the nature of this a little better, we focus our attention on the simplest examples serving to motivate what is done with economical difference schemes and regarding to some preliminaries. The object of investigation is the heat conduction equation in the space... 

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). 

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... 

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 difference approximation of every auxiliary problem from collection (55) through the use fo the simplest two-layer scheme with weights leads to an additive scheme. If either of the auxiliary schemes with the number a is economical, then. so is the resulting difference scheme. 

---