1549301742-The_Theory_of_Difference_Schemes__Samarskii

572 Economical Difference Schen1es for Multidhnensional Proble1ns

Example 3 As a matter of experience, the intention is to use a higher-

accuracy scheme similar to ( 12) being used for the heat conduction equa-

tion. As we have stated in Section l, the factorized scheme

(27)

where

(28)

x E wh, 0 < t = nr,

Yl,h=p(t), t=nr, y(x,0)=u 0 (x), xEwh,

Acx Y = y.,, ,v 0: x Ct ,

1 h^2

(} = - - _Q'._

°' 2 12 '

Cl'=l,2,

generates an approximation of O(r^2 + lhl^4 ).
Also, the scheme so constructed is equivalent to the alternating di-
rection scheme ( 46)-( 4 7) fr01n Section 1. Among other schen1es, the users
prefer two alternating direction schemes, either of which is equivalent to
scheme (27).

The first scheme:

Every equation refers to two-layer schemes with a con1mon canonical fonn

(29)

(E - ()2 T A ) 2 y - y = ()2 A 2 Y' -

T

h2

Xcx =

1

;, ct= 1,2.