1549301742-The_Theory_of_Difference_Schemes__Samarskii

626 Econmnical Difference Schemes for Multidimensional Problems

will be given special investigation, where u(t) is a solution of the Cauchy

problen1 (51) and v(c>)(t), CY = 1, 2, ... , J!, is a solution of problen1 (55)-

(56). By substituting here v(o:)(t) = z(o:)(t) + uJ+l, uJ+l = u(ij+ 1 ), c~ =

2, 3,. .. , p and v(l) = z 1 (t) + u(t) into (55)-(56) we are led to

dzc °' l

clt + Ao:(t) v(o:)(t) = 1/Jo:(t), tj < t < tj+l, 1 <CY< p,

Z( J) ( t j) = Z(p) ( t j) l j = 1, 2, ,. 1 Z j ( 0) = (^0 1)
z(o:)(tj) = z(o:-l)(tj+l), j = 0, l,,," CY= 2,3,. .. ,p,
z(tj+r) = z(p)(tj+r),
where
From such reasoning it seerns clear that
vVith the aid of the relations uj +l = u(t )+0( T)' valid for any CY = 2, 3,, '' 'p
on the whole segment t E [tj, tj+iL we finally get
where 150: 1 is, as usual, Kronecker's delta, By the sarne token,
'
p 0 p p d
L 1/J c> = L f o:(i) ~ L Ao:(i) lt(i) ~ d~ = (^0 1)
o:=l o:=I o:=l
yielding
p
V' = L v·: = 0( r).
o:=l