150 Homogeneous Difference Schemes
violated, thus causing the appearance of the extra heat source (for q < 0)
or sink (for q > 0) at the point x = (.
We call various schemes, in which conservation laws fail to be true,
nonconservative or disbalanced.
The above example shows that in designing difference schemes it is
very desirable to reproduce the appropriate conservative law on a grid.
The schernes with this property are said to be conservative. In subsequent
sections the general method for constructing conservative sche1nes, which
are convergent in the class of discontinuity coefficients, will be appreciated.
Before we undertake the complete description of this method, it is worth
noting two things.
Quite often, in practical implementations without concern of theo-
retical estimations of the desired quality of a scheme, the convergence of
the scheme is verified by experiments in which the grids are successively
refined. Sometimes this approach may cause erroneous conclusions on con-
vergence of the scheme on account of some nearness between a solution of
the difference problem and some limiting function u( x) during the course
of successive grid refinement. In the above example the function u(x) may
deviate, generally speaking, from the solution u(x) of the original problem
as large as we like. That is why the method of successive grid refinement is
employed with caution. Anyhow it cannot exclude theoretical investigations
at least in model problems.
The method of test functions is quite applicable in verifying conver-
gence and determining the order of accuracy and is stipulated by a proper
choice of the function U( x). Such a function is free to be chosen in any
convenient way so as to provide the validity of the continuity conditions
at every discontinuity point of coefficients. By inserting it in equation ( 1)
of Section 1 we are led to the right-hand side .f = (kU')' - qU and the
boundary values μ 1 = U(O) and p 2 = U(l ). The solution of such a problem
relies on scheme (4) of Section 1 and then the difference solution will be
compared with a known function U(x) on various grids.
One more thing is worth noting here. It would be erroneous to think
that every scheme, which is convergent in the case of srnooth coefficients,
is obliged to converge in the case of discontinuity coefficients. Further
explorations are connected with a family of schemes converging in the class
of discontinuity coefficients, thus expanding possibilities. Let us stress that
in the sequel we will deal with such schemes only.
- The integro-interpolational method for constructing homogeneous dif-
ference schemes. Various physical processes (heat conduction or diffusion,
vibrations, gas dynamics, etc.) are well-characterized by some integral
