1549301742-The_Theory_of_Difference_Schemes__Samarskii

474 Homogeneous Difference Schernes for Tirne-Dependent Equations

which in combination with ( 37) leads to

117)1]2 = h 7)~+1 + (1 - Xn+1) '/)~+2

thereby justifying that

lhlJ = O(Vh + Tmu) for any scheme (7),

117)1] = O(h^312 + Tmu) for the best scheme with coefficient (32).

Having stipulated the same conditions (23), (24), (30) and (31) as

before, schen1e (7) converges uuifonnly with the rate O(Vh + rmu):

This esti1nate can be i1nproved for the forward difference scheme with

CJ = 1 by means of the maximum principle and the method of extraction of

"stationary nonhomogeneities", what amounts to setting

where wj is a solution to the equation

0

( 43) A w = ·if = 'l)x ,

so that

llwllc < _.!:._ (1, hi],

c1

1

llwrllc < - (1, hrll ·

c1

Here vj needs to be recovered fro1n the relations

In this regard, the .maxi1num principle with regard to equation ( 44) yields

J

llvj+^1 llc < llv^0 llc +I: r (ll·vJ•J'llc + llw(llc) ·

j'=l

By inserting here the esti1nates for llwllc, llwtllc and llV^1 *llc we arrive at
llvjllc = O(r + h) and, hence, llzjllc = O(r + h). For the sche1ne with
the coefficient specified by (32), it retains the order of accuracy in the
class of discontinuous coefficients as occurred before in the stationary case:
llzjllc = O(h2 + r).