  This paper deals with the numerical integration of well-posed multiscale
systems of ODEs or evolutionary PDEs. As these systems appear naturally in
engineering problems, time-subcycling techniques are widely used every day to
improve computational efficiency. These methods rely on a decomposition of the
vector field in a fast part and a slow part and take advantage of that
decomposition. This way, if an unconditionnally stable (semi-)implicit scheme
cannot be easily implemented, one can integrate the fast equations with a much
smaller time step than that of the slow equations, instead of having to
integrate the whole system with a very small time-step to ensure stability.
Then, one can build a numerical integrator using a standard composition method,
such as a Lie or a Strang formula for example. Such methods are primarily
designed to be convergent in short-time to the solution of the original
problems. However, their longtime behavior rises interesting questions, the
answers to which are not very well known. In particular, when the solutions of
the problems converge in time to an asymptotic equilibrium state, the question
of the asymptotic accuracy of the numerical longtime limit of the schemes as
well as that of the rate of convergence is certainly of interest. In this
context, the asymptotic error is defined as the difference between the exact
and numerical asymptotic states. The goal of this paper is to apply that kind
of numerical methods based on splitting schemes with subcycling to some simple
examples of evolutionary ODEs and PDEs that have attractive equilibrium states,
to address the aforementioned questions of asymptotic accuracy, to perform a
rigorous analysis, and to compare them with their counterparts without
subcycling. Our analysis is developed on simple linear ODE and PDE toy-models
and is illustrated with several numerical experiments on these toy-models as
well as on more complex systems. Lie and
