1549301742-The_Theory_of_Difference_Schemes__Samarskii

The su1n1narizecl approxiination inethocl^593

leading to additive schemes. The describing sche1nes of this type will be

studied in more detail in Section 10, but a great deal of work still needs

to be done in adopting those ideas. It is worth mentioning here their main

pee uliari tie;o;:

• the passage frmn the jth layer tot.he (j+ 1 )th layer can be perforn1ed
  through the use of a sequence of the usual (two-point, three-point,
  etc.) schemes;


• the error of approximation provided by an acldi ti ve scheme is ta.ken
  to be a sun1 of the residuals of all auxiliary schemes, that is, any
  such scheme generates a summarized approximation.
  In this regard, we should take into account that auxiliary schemes are
  not obliged to approxinrnte the original proble1n; the approximation here
  is ensured by su1nrnarizing all the residuals obtained.
  In Chapter 2 we ca1ne across the necessity of generalizing the notion
  of approxin1ation in the real ;o;ituations when a difference scheme cannot
  provide on the grid w h local approximations with a desired order in the
  nonn of the space C, but it does the san1e in one of the negative nonns,
  that is, in a certain sen;o;e of summarizing.
  Likewise, it may happen that a sche1ne on the grid W 7 cannot provide
  local approxin1ations in t, but at the final stage the approxin1ation will be
  achieved once we bring together the residuals over several time layers. The
  notion of su1nn1arized approximation needs certain clarification. It seems
  worthwhile giving sin1ple examples.

Example 1 The Cauchy problem, being the most familiar one, comes
first:
du
-+aH=O dt ) t > 0 , ·u ( 0) = tt 0.

Common practice involves for solving it the difference scheme

(3)

yj+l/2 - yJ.

-----+ a1YJ = 0'

T

j = 0, 1, 2 .... '

yi+l - yj+l/2.

------+ Cl2y1+1/2 = 0'

T

which consists of two explicit schemes with residuals l/! 1 and ~; 2 , respectively.
Within more compact notations

yi+1/2 = z1+l/2 _ii,