1549301742-The_Theory_of_Difference_Schemes__Samarskii

628 Eco1101nical Differeuce Schernes for Multidin1ensio11al Proble111s

This chain can be associated with the representation of the operator A as

a sum

where A^1 = { 0.5 Aa

a 0.5A2p-a+1

for 1 < a < p,

for p < a < 2 p.

The proble1n we have posed above is of second-order accuracy in r:

11 vj - uJ 11 = 0( T2) '

provided that some smoothness property of the initial vector u 0 of the type

llA;Ap U 0 II< M, et, /3 = 1,2,. .. ,p,

holds together with the smoothness property of the operators Aa ( t) in t.

In such a way, the procedure of solving problem (51) reduces to a

sequence of sin1pler problems (55)-(57). whose solution can be obtained by

means of exact or approximate methods. In particular, the finite difference

method suits us perfectly for doing so. If the operators Aa are pairwise

com1nutative, the accuracy of an approximate method available for solv-

ing proble1n (51) depends on how well we are able to solve every auxiliary

problem with the number et frmn sequence (55). The above exploration is

still valid for the case of the homogeneous boundary conditions. In dealing

with the nonho1nogeneous boundary conditions the accuracy of the com-

posite Cauchy proble1n (.55 )-(57) depends significantly on the possible ways

of specifying the boundary conditions for v( a). The same remark applies

equally well to difference analogs of problem (55)-(57).

The difference approximation of every auxiliary problen1 from collec-

tion (55) through the use fo the simplest two-layer scheme with weights

leads to an additive scheme. If either of the auxiliary sche1nes with the

number LY is economical, then so is the resulting difference sche1ne.

Remark The accurate account of error .~h = Yh - u.h cCtu be done 1n
a nu1nber of different ways. In concluding Section 11 the usual way of
proper evaluation of the error zh was recommended for an additive scheme.
Another way of proceeding is connected with the triangle inequality
II zh II = II Yh - u.h II < II Yh - vh II +II vh - uh II'
where v is a solution of the locally one-dimensional proble1n (53)-(54) or
( 55 )-(57). Fron1 such reasoning ir seen1s clear that the further estimation
of the error zh an1ounts to evaluating the proxi1nity between Yh and vh, vh
and uh. Some progress in such matters can be achieved by the subsidiary
information about the s1noothness of the functions u. and v, thus causing
some cumbersome exposition in connection with more a detailed exploration
of the properties of the solution v of the composite Cauchy problem (53)-
(54) 01' (55)-(57).