1549301742-The_Theory_of_Difference_Schemes__Samarskii

Homogeneous difference schemes for hyperbolic equations 505

For the weighted scheme (20) these estimates are ensured by (J > (J, and

lail < c3a.

\iVe shall need yet, among other things, some modification of the well-

known estimates on an equidistant grid (see Chapter 2, Section 4), taking

on an arbitrary nonequidistant grid the fonn

(2:3)

Since D is a self-adjoint operator and

D = E + (J r^2 A = E + ( (J - (J,) r^2 A+ (J' r^2 A

c: 2 1 - E

>E+O.or A-llAllA>c:E,

we obtain

1

D-^1 < - E and

E

1

llVilln-^1 < ft II Vi II·

As usual, we may atte1npt the solution z of problem (11)-(13) as a

sum z = v + w with the members v and w satisfying the conditions

(24) w 1 t =Awl")+ ·i/J*, w(,c, U) = 0,

wt(x, 0) = IJ(x), w 0 = wN = 0.

Putting these 'together with (21)-(23) we deduce for v and w that

( 211 )

(22^1 ) llwj+^1 llc < ~ (11/Jlln + E^7 111/i*kll) ,

where 11/Jll~ = ((E+(Jr^2 A)IJ,IJ)* = ll/Jll^2 +(Jr^2 (a,IJ;J.