1549301742-The_Theory_of_Difference_Schemes__Samarskii

564 Econo111ical Difference Sche1nes for Multicliinensional Proble1ns

which are i1nmediate implications of the chain of the relations

Esti1nate (56) serves to 1notivate that scheme (51 )-(52) and one of its equiv-

alent sche111es, na1nely sche1ne (46)-(47) converge in the space HA with the

rate O(r^2 + lhl^4 ).

Re1nark By analogy with the preceding section it is possible to evaluate

the errors of approximation for either of the equations ( 48) such as

"/, _ C 1 1l -- B 2 1t n+1 ( ) n

'h - + 1 - 0"1 cp.

T

This can be done using fonnula ( 49) for the intern1ecliate value fl of artificial

character

U = 0" 1 B2un+l + (1 - 0" 1 ) C21tn.

Upon substituting this fictitious value into the above formula for 1/J 1 we find

that

thereby clarifying that the quantity v'i 0" 1 coincides with the error of approx-

in1ation for the factorized schen1e ( 51) that is known to us from fonnula

(53).

By virtue of the relation

we can be pretty sure that every of the equations ( 48) generates an approx-
imation of O(lhl^4 + r^2 ).

9.2 ECONOMICAL FACTORIZED SCHEMES

1. Schemes with factorized operator. VVe now consider the two-layer dif-
   ference scheme