1549301742-The_Theory_of_Difference_Schemes__Samarskii

586 Economical Difference Sche1nes fm· Multidimensional Problems

whose canonical form is

( 58)

y = f.-l for x E /h, t E W 7 ,

y(x,O) = u 0 (x), Yt(x,O) = ii 0 (x), x E wh,

where i:1 0 (x) = Lu 0 + f(x,O).

The following algorith111 is perfonned for the recovery of y yj + l

from scheme (58):

(E+rR 1 )w( 1 )=F^8 , s=l,2, ... ,n,

1'

w(l)=IT(E+rR,13)/-t; for a; 1 =0,l 1 ,

13=2

(E+rRa)w(a) =w(a-l)' Cl'= 2,3, ... ,p, s = 1,2, ... ,n,

p

II for Cl'=2,3, ... ,p-1,

As can readily be observed, the components w(a) can be found inde-

pendently. The primary scheme (56) becomes stable by merely setting

(} = E = const >^0.

The operators Ra are positive and pairwi~e commutativ<_J assuring us
of the validity of the inequalities Qp > 0 and R > R, where R = R + rQp
is the regularizer of scheme (58). This supports the view that schen1e (58)
0
is absolutely stable in the space r2 h.
Let y be a solution of problem (58) and u be a solution of the original
problem (50) Upon substituting y = z + u into (58) we establish for the