1549301742-The_Theory_of_Difference_Schemes__Samarskii

618 Econmnical Difference Schemes for Multidimensional Problems

where C13 are linear operators acting in a vector normed space H h. 'Ne shall

need yet n-1 initial vectors y(O) = y 0 , y(r) = Y 1 , ... , y((n - 2) r) = Yn-2

for its nun1erical solution.

By the n-layer composite scheme of period rn (of order m) we

generally n1ean a system of differential equatioins with operator coefficients

m n-2

(41) L Ca(3(tj)y(tj +,Br)= L Da(3(tj)y(tj -,Br)+fa(tj),

(3=1 f3=0

cx=l,2, ... ,ni,
with known initial values y(k r), k = U, 1, ... , n - 2. Here t 1 takes on the
values
t 1 = (n- l)r+kmr, k = 0, 1, ... I
and the total number of layers is equal to the amount of the initial condi-
tions.
vVith knowledge of the values Yj-,'3• /3 = 0, l, ... , II - L, where tj =
(ni + n - l) r, it is possible to find y(t 1 + 1n r) = YJ+m in the process of
solving a system of equations with the operator 1natrix C = ( Ca(3) of size
mx m.
Several particular cases will be given special investigation. For m = 1
the composite scheme (41) falls within the category of standard n-layer
schemes. For n = 2 the describing sche1ne is tern1ed a two-layer compos-
ite scheme of period ni

m

( 42) L Ca(3(tj) y(tj +,Br)= Dao y(tj) + fa(tj),

(3= I

y(O) = Yo·

If for the co111posite scheme ( 41) the error of approxi1nation 1jJ is adopted as
a sum of the residuals ·If; a of separate equations, that is, 4' = 4J 1 + · · · + ~'m,
the composite sche1ne (41) is called an additive schen1e.
By replacing r by r/1n scheme (42) admits an alternative fonn

m
(42') LCa(3(tj)y(tj+,6r/m)=DaoY(ij)+fa(tj),
1~=1

ex= 1,2, ... ,1n.