Asymptotic behaviour of splitting schemes involving time-subcycling techniques

This paper deals with the numerical integration of well-posed multiscale systems of ODEs or evolutionary partial differential equations (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 unconditionally 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 behaviour 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.