1549301742-The_Theory_of_Difference_Schemes__Samarskii

Econo1nical factorized schemes 577

holds.

In the case of a variable operator A= A(t) it is required, in addition,

that A( t) is Lipshitz continuous int, allowing the operator R to be constant.

In the situation when the operators R 0 , are pairwise commutative, the

stability of the primary scheme implies that of the factorized sche1ne (39),

since Qp > 0, B = E + 2 r^2 Ql' > E and R. = R + T Qp, giving

• 1 + c
  R >
  4

A.

Let us stress that the well-founded choice of the regularizer R is in

complete agreement with an approved principle governing what can happen

and depends on the range of situations to be considered. This is certainly

so with two-layer and three-layer sche1nes and, therefore, one and the same

regularizer R could be useful and perceived to be useful for different oper-

ators A.

Example 4 By having recourse to problem (23) associated with the heat

conduction equation with variable coefficients for the same choice of the

operators A, R 1 and R 2 as in Example 1 for the two-layer economical

scheme (24) we concentrate on the primary scheme (36)

yo t + T^2 R Ytt +A Y = cp,

which generates an approximation of order 2 and possesses the residual

·,P = O(r^2 + lhl^2 ). This schen1e is stable for u > (1 + s)/4. Observe that

the factorized scheme of the type (39) with the inembers

turns out to be absolutely stable for u > (1 + s)/4, s > 0. This is clue to

the fact that the operators R 1 and R 2 are commuting.

Also, the operator A(t) will be Lipshitz continuous if we agree to

consider

l(au) 1 I < c 3 a 0 ,, c 3 = const > 0, CY= 1, 2.

S. mce every operator B °' = E +^2 T R °' is. " econom1ca. l" , t^1 le same property
holds true for scheme (39).
In an attempt to solve the system of difference equations