1549301742-The_Theory_of_Difference_Schemes__Samarskii

624 Eco1101nical Differeuce Schemes for Multidimensional Problems

Also, the operator A will be taken to be

(52)

where linear operators Aa are so chosen as to provide the relation

p

n D(Aa) = D(A),

a:=:I

making it possible to reduce the solution of the Cauchy problem (51) to

solving successively the Cauchy problems of the same type, but with oper-

ators Aa standing in place of the operator A. We confine ourselves here to

two possible ways of such a reduction.

For later use, it will be sensible to introduce on the segn1ent 0 < t < t 0

a grid w 7 = {tj = jr, j 0, 1, ... ,j 0 } with step T and to atten1pt the

function f in the form

p

f = 2= fa.

The first reduction (for more detail see Section 3). The object of

investigation is a chain of the equations

(53) et=l,2, ... ,p,

with the supplementary initial conditions

(54) j=l,2,. .. ,

j=O,l, .. ., et=2,3, ... ,p.

The function v(tj+i) = v(p)(tj+i) refers to a solution of this problen1
fort= tj+J · In the general case we might have

for all j = 1,2, ...