1549301742-The_Theory_of_Difference_Schemes__Samarskii

Economical factorized sche1nes 571

can be solved using Gaussian elirnination along the rows for Cl' = l as well

as along the columns for Cl' = 2.

Under such an approach the factorized scheme is of no less than first-

order accuracy in T. In a similar way an economical factorized scheme can

be designed in the p-din1ensional case when

( 231 )

and the operator Lcx is specified by formula (23). If so, we might agree to

consider

}J

B = II ( E + T Rcx) ,

cx=l

Here the operators Rex are of the same structure as was chosen for p = 2.

Example 2 The state1nent of the first boundary-value problem for the

parabolic equation with mixed derivatives in the parallelepiped Go = {O <

X ex < lex , Cl' = 1 , 2,... , p} is

(26)

au

• = L 1l + f(x, t), ulr = μ(x, t), u(x, 0) = u 0 (x),
  at
  I'
  L1t= L
  cx,;3=!
  }J }J }J
  0 < C 1 L ~~ < L kexμ(x,t)~ex~/3 < c 2 L ~~.
  ex= l ex,;3=1 ex=l

As the operators Rcx involved, we take once again the operators specified
by formulas (25) that are self-adjoint, positive (for (} > 0) and pairwise
0 -
commutative in the space 0, h, since Go is a parallelepiped. The sche1ne
with these members is stable for (} > 0.5. As far as Bex = E + rRex,
Cl' = 1, 2, ... , p, are three-point difference operators with constant coeffi-
cients, the possible follow up is the algorithm suggested in Example 1 for
determination of yj+l on the basis of yj. We will not pursue analysis of
this: the ideas needed to do so have been covered.