1549301742-The_Theory_of_Difference_Schemes__Samarskii

598 Economical Difference Schemes for Multidimensional Problems

For example, the weighted scheme

suits us for doing so. Here Aa ,...., La and (}a is an arbitrary parameter.

The governing equation Pav(a) = 0 is approximated by scheme (8) in

the usual sense so that

(9)

tends to zero in some suitable norn1 as T--+ 0 and ha --+ 0.

The difference equations (8) constitute what is called an additive

scheme. Indeed, let 1/Ja = IIauj+a/p be the i·esiduals of the same scheme

(8) with the number Cl' attached.

Ananging 1/J a as a sum

( 10)

and taking into account that

we deduce that

0

·1.!J ' (l' = (P (l' u)J+^1!^2

and 11·1/J:11 --+ 0 as T --+ 0, h --+ 0, where 11 · II is son1e suitable norm on
the space of all grid functions given on the grid w h. It follows from the
foregoing that

that is,

p

111/J II= II L 1/Ja II -+ 0 as T--+ 0, lhl--+O,

O! ::0 1

scheme (8) generates a summarized approximation if either

of the schemes (8) with the number Cl' approxi1nates the

corresponding equation (6) in the usual sense.