1549301742-The_Theory_of_Difference_Schemes__Samarskii

582 Economical Difference Sche1nes for Multidiinensional Proble1ns

Having replaced the operator E + r^2 ( R1 + R 2 ) by the factorized op-

erator (E + r^2 R 1 ) (E + r^2 R2), where R<,y = -(} Y:rc""", Cl'= 1, 2, we obtain

under such an approach the economical difference scheme

YI, = μ, y(x, 0) = u 0 (a:), Yt(x, 0) = u 0 (x),

with u 0 = u 0 + 0 .5 r( Lit + f) It =Ll incorporated. This scheme is of second-

order accuracy in T and lhl. vVhat is more, it is absolutely stable and

appears prefera.ble in practical implen1entations.

5. Economical schemes for a syste1n of equations of parabolic and hyperbolic
   types. Let G = {O < xn < lcr, Cl'= 1,2, ... ,p} be a parallelepiped in the
   space RP,

Qr = (; x [O < t < T] , Qr = G x (0 < t < T]

and k = (ko:/3) = (k~^1 3), s, 1n = 1, 2, ... , n, be a matrix of size p x p with

square blocks of size n x n satisfying the condition of symmetry

( 46) k~,;(x, t) = e;;(x, t) for all (x, t) E Qr

as well as the condition of positive definiteness

n}J n p n}J

(47) c1LL(~~)

2

< L L k~

1

~(x,t)~~,~~<c2LL(~~)^2 ,

s=l a=l s)rn.=1 cx,/3=1 s=l a=!

where c- 1 and c 2 are positive constants and ~°' = ( (;, ... , ~~, ... , ~~) is an
arbitrary real vector. The positive definiteness of the matrix k is equivalent
to being strongly elliptic of the operator L with the values

( 48)

I'

Lu = L L 0 • 1 _, u,

CY,/3=1

where u = ('u^1 , ... , u^8 , ••• , un) is a vector of order n. The meaning of this
property is that we should have

( 49) c l (-L(^0 lu J u) <(-Lu - J u) )