Preface vthe fourth stage the computational procedures need to be investigated and
implemented. The fifth stage focuses on the review of their delivery and
necessary corrections to the approved mathematical model governing what
can happen.
It may turn out that the inathematical inodel is too rough, n1eaning
numerical results of computations are not consistent with physical exper-
iments, or the model is extre1nely cumbers01ne for everyday use and its
solution can be obtained with a prescribed accuracy on the basis of sin1-
pler tnodels. Then the same work should be started all over again and the
remaining stages should be repeated once again.
The stages of numerical experimen(.s for theoretical investigations of
physical problems were explained above. Greatest attention is being paid
to a new technology for research with rather complicated 1nathen1atical
models.
Classical n1ethods of mathematical physics are employed at the first
stage. Numerous physical problems lead to mathematical models having
no advanced n1ethocls for solving them. Quite often in practice, the user is
forced to solve such nonlinear proble1ns of mat.hematical physics for which
even the theorems of existence and uniqueness have not yet been proven
and some relevant issues are still open.
For the m01nent, we are interested only in the second stage of nu-
merical experiments. A computational algorithm usually means a sequence
of logic and arithmetic operations that enables us to solve a problem. A
computational algorithm is developed to solve the problem with any degree
of accuracy E > U in a finite number of operations Q(c:). This is the gen-
eral requirement, thus raising rnany mathen1atical (jUestions. vVhat does
the expression "with any degree of accuracy" mean? A possibility, at least
in principle, of obtaining a solution with a prescribed accuracy should be
substantiated. However, smnetimes it n1ay happen that Q( E) is finite for
son1e E, but it is so great that in practice it is unrealistic to produce a so-
lution with such a degree of accuracy. For any problem an infinite nun1ber
of various computational algorithms may be constructed possessing, for ex-
atnple, similar a.sy1nptotic properties, so that Q( E) will be of the same order
in E as E -+ 0. Not all such tnethods should be involved, but only those
suitable for computational restrictions. Naturally, the methods should be
used that require the minimal execution time for solving the proble1n with
a prescribed accuracy. The execution time must be reasonable, that is,
measurable in minutes or hours if such calculations are encountered very
rarely. Of course, the cmnputing process must be as inexpensive as possible.
The time for solving the problem depends on the algorithn1s, quality of the
program and c01nputer performance. It is difficult to evaluate the latter