1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternating direction n1ethocl 559

or

Sumn1ing up over k = 0, 1, ... , n and inserting z^0 = 0, we find that

( 43) llzn+lll(l l = II (E + 0.5 T Az(t11+1)) zn+l II

< ~ t T (111/J~ II+ II~; II) ,

k=O

where

( 44)

Observe that estimate (43) remains unchanged ifthe nonn llzll(l) is replaced

either by the nonn II z II or by the norm

by virtue of the relations

This type of situation is covered by the following assertion.

Theore1n 3 Scheme (33)-(34) is absolutely stable (for any h 1 , h 2 and r)

and converges with the rate O(lhl^2 )+r^2 in tl1e norm 11 · ll(l) given by formula

( 44) under the rondi tions which ,g·uarantee a second-order approximation

on a solution u = u(J:, t) of proble111 (:32).

Recall that a second-order accuracy of sche1ne (33)-(34) is ensured

by making a special choice of the boundary conditions for the intermediate

value y = y such as

y =fl for

It is possible to demonstrate that the accuracy O(lhl^2 ) + r^2 of this
scheme remains unchanged if we might agree to consider

( 45)

μn + μn+l

y=

2