1549301742-The_Theory_of_Difference_Schemes__Samarskii

Homogeneous difference schemes for hyperbolic equations 503

Following the procedures of Chapter 3, Section 4 and taking into ac-

count that the second partial derivative ff^2 1l/ ot" is continuous on the line of

discontinuity x = ~ of the functions k( x, t) and f ( x, t), we deduce through

such an analysis that

Because of this, formulas ( 13)-(15) are an immediate i1nplication of formula

(17) with the preceding expansion involved. From representation (14) it

see1ns clear that

17; = O(h^2 2

1 +r ), 17t; = O(h

(^2) + r").
' '
( 1 g)

4. Stability and convergence. No restrictions a.re ma.de regarding the
   smoothness of the coefficients and the solution in the further esti1nation
   of the accuracy of scheme (7)-(9). This can be done using various a priori
   estimates for the operator-difference three-layer scheme

(20) D Zft +A z = ij;(t)' t = iT > 0'

z(O) = 0, zt(O) = JJ.

Here D and A stand for linear operators in a Hilbert space H, z(t) and tf;(t)
refer to abstract functions of the argument t E w 7 with the values in the
space H and vis an element of the space H (for more detail see Chapter 6).
0
In preparation for this, H = D is the space of all grid functions given on
the grid wh and vanishing on the boundary at the points x = 0 and x = 1.
The usual inner products are defined to be

N-1

(z, v)* = L ziv;fii,

i=l

N-1

(z, v) = L zivihi,

i=l

N
(z, v] = L zivihi.
i=l
In the general setting three types of suitable norms are in common
usage:

llzllc = xEwh n1~X lz(x)I, II z II=~*, llzllA = V(Az, z) ·