Adaptive time-scale decomposition for multiscale systems

Multiscale systems are a class of systems where accurate modeling requires the inclusion of multiple temporal and/or spatial scales. Numerical simulation of multiscale systems is often computationally intractable. The modeling goal frequently requires a high resolution simulation and this implies a high order model. A high order model is computationally challenging on its own, but can be impossible with current hardware and algorithms when combined with the multiscale property.; Multiple time-step (MTS) integration is a potential solution to the temporal multiscale problem. In an MTS integrator the component functions of the system are grouped into distinct sets by time-scale and each group is evaluated with a time-step of appropriate size. This type of integrator was originally developed for free body mechanics simulations, specifically molecular dynamics. The reason they have not been used in a broad class of multiscale systems is that there has not been a general way to separate the component functions into groups.; This work is focused on the development of a decomposition scheme for oscillatory systems. The basis of the method is time-frequency analysis of the time series of the component functions of the ODE; specifically, the Short-Time Fourier Transform (STFT) is used. From the frequency spectrum a characteristic frequency is derived that describes the time-scale of the component function. The set of characteristic frequencies are used in a set partitioning problem to split the component functions into fast and slow subsets. The methods are studied both with and without the use of a high-pass prefilter based on the empirical mode decomposition.; These methods are evaluated by testing on three example systems. The methods have been shown to partition and repartition the system correctly as the system evolves in time. The convergence and time-scale separation properties of the empirical mode decomposition (EMD) for oscillatory signals has been proven under certain assumptions. And the use of the EMD as a high-pass prefilter has been shown to improve estimates of the characteristic frequencies in the presence of low-frequency background signals. The effectiveness of these methods for moderately stiff systems has been demonstrated.