1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable two-layer schernes 423

12. Example. In order to apply the general stability theory for particular
    difference schemes, one needs to perform several manipulations:
    
    
    • to reduce a two-layer scheme to the canonical form (1), that is, to
      specify the operators A and B involved;
    
    
    • to introduce the space H h of all grid functions and reveal the de-
      sirable properties of the operators A and B such as positiveness,
      self-adjointness, etc. as operators acting in the space H h;
    
    
    
    

• to verify whether or not the scheme belongs to the primary family
  of schemes as well as the fulfihnent of sufficient stability conditions
  (64) or (65);


• if the preceding conditions are satisfied, the scheme at hand is stable
  and a priori estimates hold for it such as, for example, (69) and (70).
  The first step within this framework is to reduce a scheme to the
  canonical form, but the above sufficient conditions provide a real possibility
  of writing stable difference schemes in11nediately in canonical form.
  We cite here only one possible example which helps motivate what is
  done. Those ideas are connected with the heat conduction equation

OU

at

EJ21l

fJx2,

u(x, 0) = u 0 (x),

O<x<l, t > 0 l

u(O,t) = u(l,t) = 0,

and associated asymmetric scheme which is given on the grid

and asquires the form

1

(71) Yi,j+l = w +CY (CYYi-1,j+l +(l-CY)Yi-1,j +Yi+l,J-(2-w-CY)Yi,J,

where w = h^2 /r and CY is a parameter.
\Ve do follow established practice in a step-by-step fashion. 1) Re-
duction of the sche1ne which interests us to the canonical form. Denoting
Yi,J =Yi and Yi,J+l =Yi we first rewrite (71) as

(72) (w+CY)Yi=o:fJi-l+(l-CY)Yi-l+Yi+ 1 -(2-w-CY)Yi·