1549301742-The_Theory_of_Difference_Schemes__Samarskii

616 Econmnical Difference Sche1nes for Multidiinensional Proble1ns

since the quantity T/(o:) is recovered from equation (38) everywhere on wh +

ih , o:. On the other hand, 1/J* 0: = 0( h 0:^2 +r) at the regular nodes of the grid w h

and i/J: = O(h; +r) at the irregular nodes. Therefore, h^2 ll~llc• = O(h^2 +r)

and 11~11 o = O(h^2 + r), so that estimate (40) in such a setting gives

c

since T)j = 0 for all j = U, 1, ... ,j 0.

Observe that the stability with respect to the right-hand side and

boundary conditions implies that the 1noments t: and t:* can be arbitrarily

taken from the interval (tj, tj+ 1 ).

9. LOS for equations with variable coefficient.s. One way of covering equa-
   tions with variable coefficients is connected with possible constructions of
   locally one-din1ensional sche1nes and the n1ain ideas adopted for problem
   (15). It sufficies to point out only the necessary changes in the formulas for
   the operators Lo: and AL>' which will be used in the sequel, and then bear in
   rnind that any locally one-dimensional scheme can al ways be written in the
   form (21)-(23). Several examples add interest and help in understanding.

1. A linear equation of parabolic type.
   Let in the state1nent of proble111 (15) involves

Minor changes in the con1plete posing of problem (21)-(23) are based on

the fon11nla for the difference operator Ao: acting in accordance with the

rule

• 0 < c j ~ Cl 0: < C2 ' t = t j + 1 I 2.

A second-order approximation provided by the difference operator Ao: on a
regular pattern

is ensured by a proper choice of the coefficient ao:. This can be clone using,
for exarnple,

a CY =k a (x(-n^5 0:)[1 ,~),

X(-o.^5 0:) _ - (x ll · · ·' x ci;-ll x a - 051, · va, x o·+I' · · ·' x) p '