Asymptotic behavior of splitting schemes involving time-subcycling techniques

In order to integrate numerically a well-posed multiscale evolutionary problem such as a Cauchy problem for an ODE system or a PDE system, using time-subcycling techniques consists in splitting the vector field in a fast part and a slow part and taking advantage of this decomposition, for example by integrating the fast equation on a much smaller time step than the slow equation (instead of having to integrate the whole system with a very small time step to ensure stability for example). These techniques are designed to improve the computational efficiency and have been very widely used for designing schemes, that may have (at least) one component that has to be computed through an explicit scheme thus constrained by a limitation of the time step (CFL). In this paper, we study the long time behavior of such schemes, that are primarily designed to be convergent in short-time to the solution of the original problem. We develop our analysis on ODE and PDE toy-models and illustrate our results numerically on more complex systems.