1549301742-The_Theory_of_Difference_Schemes__Samarskii

580 Economical Difference Schemes for Multidimensional Problems

on which the statement of the difference proble1n is

(43) Ytt = A ( (} fJ + ( 1 - 2 (}) y + (} y) + \0( x, t) ,

t=jr, j=l,2, ... , xEwh,

yf 1 h =μ(x,t), f=ir, j>O.

where 1! 0 (x) = u 0 (x) +0.5 r(Lu 0 + f(x, 0)), A= A1 + A2 and Acxy = Yxaxa.

Assuming the primary scheme ( 43) to be stable, that is, granting

4() > 1 + c;, c; > 0, we rewrite scheme (43) in the canonical form

( E - (} T^2 A) Ytt = A y + \0 '

whose use permits us to turn to the economical factorized scheme

( 44)

Another way of proceeding is to reduce sche1ne ( 4;3) to

Having con1pleted the factorization of the operator E-(} r^2 A at the member

Yt so that

we deduce that

( 45)

Both factorized schemes ( 44) and ( 45) generate second-order approxima-
tions in T for any (} and they are stable under the condition 4() > 1 + c;,
c; > 0, since the operators Rex = -Acx are self-adjoint, positive and commu-
o
tative in the space H = r2 h of all grid functions given on the grid W1z and
vanishing on the boundary ih of the grid.
In an at tern pt to recover y = yJ +i from the difference equations just. es-
tablished, we shall need the boundary conditions for an intermediate value.