1549301742-The_Theory_of_Difference_Schemes__Samarskii

454 Stability Theory of Difference Schemes

Theorem 10 If conditions (87) and (102) are satisfied, then for scheme

(101) a priori estimate (106) takes place. In particular, for scheme (101)

with D = E and y 0 = y 0 = 0 we have

(see Chapter 5, Section 6.2).

As an example consider the weighted scheme

Substituting y + CJT^2 Ytt for CJy + (1 - 2CJ)y + CJY we obtain

(107) ( E + (! T^2 A) Ytt + A y = <p '

yielding D E + CJT^2 A. The stability condition D >^1 !' r^2 A or E >

((1 + c:)/4 - CJ) r^2 A is ensured by

(! > l+c:

4

1

For the explicit scheme with CJ = 0 this implies that

2

T <

J(l+c:)llAll

Having stipulated this condition, the explicit scheme (Yti = Yxx) for

the string vibration equation is stable for r/h < 1/J(l + c:) (see Chapter

5, Section 6).

10. On regularization of difference schemes. Stability theory of difference
    schemes outlined in this chapter may be useful for the statement of a general
    principle (the regularization principle) providing with schemes of a desired
    quality, that is, stable, generating an approximation and satisfying the extra
    economy requirement of minimizing the arithmetic operations necessary in
    computer implementations of resulting difference equations.
    The economy requirement in the case of nonstationary problems in
    mathematical physics generally means that the number of arithmetic op-
    erations needed in connection with solving difference equations in passing
    from one layer to another is proportional to the total nmnber of grid nodes.