Difference schen1es for the equation of vibrations of a string 377In giving a generalized solution of the problern(43)82 u 82 u
8t2 dx = a2 OJ:20'8u
Ft(x, 0) = u 1 (x), u(O, t) = 0, u(l, t) = O,O<x<l, t>O,
u(x,0) = u 0 (x),
we rely on the scheme with weights( 44) A Y -- a^2 Yxx'with the appropriate supplementary conditions. When studying the conver-
gence of the scheme with weights, it is supposed that a solution of problem
( 43) exists and is smooth enough. It is possible under certain conditions
relating to the smoothness of the initial data.
Does the same sche1ne converge if u = u( x, t) is a generalized solution?
The grid solution of problem ( 44) turns out to converge to the generalized
solution with the rate 0( VVh + r). We do not dwell on confirming this
statement.
In trying to find a generalized solution of problem ( 43) one can come
across oscillations of the grid solution and its derivatives ("ripple"), which
essentially reduce the accuracy of a scheme. What is more, the lines of
discontinuity of derivatives spread over several grid intervals, thus causing
the difficulties in determination of proper discontinuity propagation veloc-
ity. This is the result of introducing the fictitious friction (dissipation) in
the difference approximation.
The ripple is stipulated by the fact that difference harmonics reveal
the dispersion, that is, deterrnination of a harmonic velocity depends on
its number, whereas for the differential equation all harmonics have the
sarne velocity a. In order to improve the quality of a scheme, one needs
to minirnize the dispersion. Among various schemes ( 44) with weights the
scheme relating to·O'=O' * =2-(1-2-) 12 , (^2) , I=
llT
h ,
has the minimal dispersion and allows us t.o overcome the obstacles men-
tioned above. It is of order 4, that is, 0( r^4 + h^4 ) on sufficiently smooth
solutions u = u(x, t). On nonsrnooth generalized solutions the approxima-
tion error for scheme ( 44) with the weight O' = O', is as large as for schernes
with any weight O' # O',. However, due to reduced dispersion the scheme