1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable three-layer sche111es 455

(108)

(109)

In studying the canonical form of two-layer and three-layer schemes

(E+rR)y 1 +Ay=<p, y(O)=y 0 , B=E+rR,

Byo t +r^2 Ryr 1 +Ay=<p, y(O)=y 0 , y(r)=Y1,

it has been disclosed that the operator R ( regularizator) is responsible

for stability.

( 110)

Sufficient stability conditions have now sin1plified forms

R > cr 0 A,

1

R>-A

4

or

1 1

2 rllAll

for two-layer schemes,

R >

1

• 
  
  
  
  • 4 E A for three-layer schemes.
  
  
  
  

Stability or instability of a scheme from the primary fa1nily depends
only on selection rules for the operator R. From the point of view of stability
theory the arbitrariness in the choice of the operator R is restricted by the
following requirements:

• a sche1ne should belong to the primary family, that is, B = E +TR
  for (108) and R = R* > 0 for (109);


• conditions (110) should be valid.
  To obtain a stable sche1ne of a desired quality, one is to construct
  a scheme generating an approximation of the attainable order and being
  economical, that is, it is required to solve the equations ( E + T R)f; = F for
  (108) or (B + 2rR)f; = F for (109) in a minimal nmnber of operations (in
  a certain sense).
  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. Cmnmon practice
in designing difference schemes involves the develop1nent of a scheme which
generates an approxin1ation of the attainable order and is economical. With
the indicated properties, its stability will be given special investigation.
The main idea behind regularization of difference sche1nes is that the
schemes of a desired quality should be sought in the class of stable schemes
starting from an original scheme and replacing it, by changing the operator
R, by another scheme of a desired quality belonging to the class of stable
schemes.
Many 111ocles of constructing sche1nes of a particular form can be
treated as simplest regularization modes. The canonical form of a sche1ne
is convenient not only for practical tests of stability, but also for proper