1549301742-The_Theory_of_Difference_Schemes__Samarskii

The su1n1narized approximation method 631

grid in T on the segment 0 < t < T and a grid wh in G remains unchanged

(for more detail see Chapter 5).

If G is a parallelepiped in the space RP, problem (63)-(65) can be

solved through the use of an economical factorized scheme with accuracy

O(r^2 + lhl^2 ). The design of such a scheme was made in Section 2 and it

was investigated there in full details. Applying the same procedure serves

ti 1not.ivate that, first, the operators

(66)

with fa still subject the normalization condition

are approximated successively with step T /p.

Second, the difference expresions

(67)

are ai111ed at approxin1ating to the derivative 82 tt/8t^2 with step r/p, where

u(u)=tt - j-l+a/2 , ll(o)-tt-tl - j-1 ,

( 68)

u(a) - ll(c"-1) - tl(a-2) + il(o:)

rv - --

T2 9 fJt2 '

C\'=l,2,3 for p=:3,

with the members u(-l) = ii( 2 ) = u(j-l)+^2!^3 and u(- 2 ) = u(l) = ttj-^2 /^3.
Third, the operator Lau+ fa is approximated to second order by the
homogeneous difference operator Ao:Y +'Po: on the grid wh in the space RP.
The coefficients at the member Ao: and the righ-hand side 'Po: are taken at
the moment