  Many applications of contemporary science involve multiscale dynamics, which
are typically characterized by the time and space scale separation of patterns
of motion, with fewer slowly evolving variables and much larger set of faster
evolving variables. This time-space scale separation causes direct numerical
simulation of the evolution of the dynamics to be computationally expensive,
due both to the large number of variables and the necessity to choose a small
discretization time step in order to resolve the fast components of dynamics.
In this work we propose a simple method of determining the closed model for
slow variables alone, which requires only a single computation of appropriate
statistics for the fast dynamics with a certain fixed state of the slow
variables. The method is based on the first-order Taylor expansion of the
averaged coupling term with respect to the slow variables, which can be
computed using the linear fluctuation-dissipation theorem. We show that, with
simple linear coupling in both slow and fast variables, this method produces
quite comparable statistics to what is exhibited by a complete two-scale model.
The main advantage of the method is that it applies even when the statistics of
the full multiscale model cannot be simulated due to computational complexity,
which makes it practical for real-world large scale applications.
