1549301742-The_Theory_of_Difference_Schemes__Samarskii

Economical factorized schemes

where

(u, v) = t ;· tts(x) v^8 (x) ell;, ell:= cfa: 1 ... clxp,

s=l G

P 82u

L(^0 lu -- u "u -- L..., ~ -8x2 '

CY=) CY

583

and u is an arbitrary sufficiently smooth vector-function vanishing on the

boundary r.

The problem here consists of finding a continuous in Qr solution of

the system of parabolic equations

(50)

8u

at = L u + f ( x, t) , ( x, t) E Qr ,

u=,u(x,t)forxEf, tE[O,T],

Before going further, it will be sensible to introduce in C a grid

and on the segment 0 < t < T a grid w 7 = {tj = jr, j = 0, 1, ... }.
The operator L 0 :r3 is approximated by the difference operator Aaf3 acting
in accordance with the rule

(51)

with the well-established notation

(52)

p

AU = L Aaf3 U.

a,(3=!

Along these lines, we obtain for /3 = ct

a a = (^0) • 5 (k aa + k(-lc.)) aa ·
0
The inner product in the space r2 h of all grid vector-functions given on the
grid w h and vanishing on the boundary /h is defined by
ll
( y' v) = L ( y' , L's ) ' (y', u') = L y8 (.t) u' ( .c) h 1 · · · h 7 , •
s=l J..'EU)h