1549301742-The_Theory_of_Difference_Schemes__Samarskii

326 Difference Schemes with Constant Coefficients

Omitting the terms O(h^2 ) in (65) and involving the equation u = u" + f

insight, we deduce that sche1ne ( 64) provides an approxi1nation of 0( h^4 +r^2 )

with the n1embers

Z/) = j + /2 h 2 ( j II + j ) ,

In trying to recover f) fro1n (63)-(64) we obtain the three-point equations

(66)
with the right-hand sides -F; depending on y, y, zp and the usual boundary
conditions for i = 0 and i = N. This problem can be solved by the right
elimination method. In the process of computing the values of y and y on
two previous layers should be placed in the storage. In the case of two-layer
schemes it suffices to save merely one preceding layer.
Stability of three-layer schemes will be established in Chapter 6. For
the purposes of current section we confine ourselves to sufficient stability
conditions for the symmetric scheme (63) and scheme (64), respectively:
(J' > ~ and (J' > - ~. In just the same way as we did for two-layer schemes
the difference boundary conditions with a highly accurate approximation
can be developed for the third kind boundary conditions (50) and (54). For
the symmetric scheme (63) the boundary conditions providing an approxi-
1nation of O(h^2 + r^2 ) reduce to

with

i -- (^0) ,
; ' -- N ,
A+y =
μ2 (t)
zp+=J(l,t)+ -l-
1
I
2!
showing the new notations to be sensible ones.
On the same grounds as before, the third kind difference boundary
conditions for the nonsym1netric schen1e (64) are specified by the relations
i = 0 I
(67)
i = N.