1549301742-The_Theory_of_Difference_Schemes__Samarskii

614 Economical Difference Sche1nes for Multidimensional Problems

In that case D(P) = l/r and Theorem 3 in Chapter 4, Section 2 states

that

(36)

Fj+a/p

llvj+a/plle <II °' D lie= T llF1+a/plle

< llvj+(a-l)/plle + T II &~+a/pile·

The first sumrnation of (36) over CY = L 2, ... , p gives rise to the relation

p

llvj+Jlle < llvj lie+ T 2= II &~+c>/plle,

a7l

which is followed upon summmg up over J

estin1ate

0, 1,2,, ... ,J 0 - 1 by the

)o-l p

(37) lld"lle < 2= T 2= II &~+a/pile·

j'=O a=l

Estimate (30) for a solution of problem (21)-(23) is an im1nediate
implication of the collection of relation (34), (35), (37) if we might invoke
the arbitrariness in the choice of the number Jo.

8. Uniform convergence of LOS. This type of situation is covered by the
   following assertion.

Theorem 2 Let problem (15) possess a unique solution u = u(x, t) con-
tinuous in Qy and there exist continuous in Qt 0 derivatives

32u

[)t2 '

[J2 f

3x^2 Ct '

Then scheme (21)-(23) converges uniformly with the rate O(h^2 + r) (it is
of first-order accuracy in T and of second-order accuracy in h ), so that

J = 1, 2, ... '

where h = 1nax ha, lvf = const > 0 is independent ofr and ha.

1 .'S Ci .'Sp