1549301742-The_Theory_of_Difference_Schemes__Samarskii

488 Hom.ogeneous Difference Schemes for Time-Dependent Equations

with c(x, t) > c 1 > 0 and 0 < c 1 < k(x, t) < c 2 In view of this, the

corresponding homogeneous difference sche1ne becmnes

(76) p(x, t) Yt = A(i) y(a) + <p(x, i),

where p and <p are calculated by means of the same pattern functional

p(xi,l) = c(xi,t) or p(a: 1 ,i) = ~ (c(.1:;-0,i)+c(xi+O,l)) in the case where

c( a:, t) is discontinuous at the node x = xi.

With the detailed forms of these functionals, schen1e (76) is stable

under the constraints

1

Clo = 2

c l

rllA(i)ll'

llAll = llAll-

Because of this fact, it is unconditionallv " stable for cr -> 0.5.

Remark So far we have considered only equidistant grids int. But it is not
difficult to show that the preceding estimates for two-layer schemes remain
valid on nonequidistant grids with step Tj = tj -t j _ 1 being a function of the
subscript j. It is obvious that the grid int is rather flexible in comparison
with the grid w h as a result of refining the step Tj in the regions of the
widely varying right-hand side f(x, t), boundary values j.l 1 (t), μ 2 (t) and
the coefficient k = k(x, t) int. With knowledge of the behaviour of the
problem solution on a sparse grid, successive grid refinement will be caused
by the necessity of diminishing the step Tj in some intervals during which
the solution varies very fastly int. Any changes in the composition of the
grid w h with changes of ti in the process of calculations are connected with
indeterminate values of the function yf at new (additional) nodes of the
grid w h in view.
For example, a greater gain in accuracy in T will be achieved once
we perform parallel calculations on several grids w 7 i and w 72 by the rules
approved in Chapter 3, Section 4.
The forthcoming procedures serve to 111otivate what is clone on an
equidistant grid wh 7 , on which the representation takes place:

(77)

where CY·. ZJ and fJ·. ZJ are independent of hand T both.