1549301742-The_Theory_of_Difference_Schemes__Samarskii

The sum1narized approximation 1nethod 627

This means that the systen1 of differential equations (55)-(56) generates

an approximation of order 1 in a summarized sense to the Cauchy problern

(51) under the extra restrictions on the existence and boundedness of the

derivative A(t)d^2 u/dt^2 in some suitable nonn.

Further comparison of the solution v(t 1 ) of problem (55)-(57) with the

solution 11(tj) of the original proble1n allows to cite without proofs several

interesting re111arks.

a) Let f = 0 and a.ll the functions fu = 0. If constm1t operators An

are pairwise con1mutative: AaAμ = AμAa, o, f3 = l, 2, ... ,p, then for any

T the equality holds:

(58)

where v is a solution of problem (55)-(57) and u is a solution of problem

( 51).

When the operators AL> = AL, ( t) happen to be dependent on the tirne

t, equality (58) is still valid for commutative operators An(t') and Aμ(t"),

ex f. (3, taken at different mornents t' f. t^11 , so that

An(t') Aμ(t") = Aμ(t") An(t')' ex' f3 = 1, 2,. .. ,p'

for any t', t" E [O, t 0 ].

In this regard, we refer the readers to a few examples of Section 4 in

which equality (58) holds true for c01nn1utative operators Aa Aμ= Ao An.

b) If the operators A°'(t) and Ar 3 (t) are non-c01111nutat.ive, then esti-

mate

(59) j= 1,2, ... '

will be valid under the additional condition of smoothness:

ex,{3= 1,2, ... ,p.

Is it possible or not to in1prove the accuracy in T without essential
rnodifications of the cornposite Cauchy problen1? In an at tern pt to give
a definite answer to this question, the composite Cauchy problem (55) is
designated by the sy1nbolism

The sym111etrized problem consists of 2 p Cauchy problen1s such as

-----+ 0.5Ap-----+ 0.5Ap-1-----+ · · ·----+ 0.5A 1.