1549301742-The_Theory_of_Difference_Schemes__Samarskii

Economical factorized schemes 579

In this direction, the intention is to construct in a similar inanner the

factorized sche1ne related to another primary sche1ne

p

( 41) R = L Ra,

a=l

where the operators Rex are pairwise commutative, self-adjoint and posi-

tive. This scheme is certainly associated with the hyperbolic type equation

82 u/ 8t^2 = L1t + f. There are several mechanisms for passing from scheme

( 41) to the factorized sche1ne with the accompanying factorized operator at

the n1embers f;, Ytt or Yt. The usual way of covering this is connected with

further replacement of the operator E + r^2 R by the factorized operator

leaving us with the scheme

CJ -

( E + T-R) Ytt + A y = 'f )

which is stable only if the prin1ary scl1en1e is stable, since R - R* > R.

The resulting scheme differs from the primary one within a quantity 0( r^2 ).

Example 6 To avoid generality for which we have no real need, the

object of investigation is the equation of hyperbolic type in the rectangle

G 0 = {O < xcx < lcr, Cl'= 1, 2}. The boundary conditions of the first kind

are specified on its boundary r. The cmnplete posing of the problem is

described by

( 42)

8 •) -u

• 2 =(L1+L2)11+f(x,t),
  at

xEG 0 , tE(O,T],

ulr=μ(x,t), t>U, u(x,0)=u 0 (x),

all

ox(x,O)=u 0 (x), xEG 0.

'vVe contrived to do the necessary factorization in a number of different
ways. First, we initiated the construction of a primary weighted schen1e on
an equidistant n~ctangular grid