We introduce a new strategy for coupling the parallel in time (parareal) iterative methodology with multiscale integrators. Following the parareal framework, the algorithm computes a low-cost approximation of all slow variables in the system using an appropriate multiscale integrator, which is refined using parallel fine scale integrations. Convergence is obtained using an alignment algorithm for fast phase-like variables. The method may be used either to enhance the accuracy and range of applicability of the multiscale method in approximating only the slow variables, or to resolve all the state variables. The numerical scheme does not require that the system is split into slow and fast coordinates. Moreover, the dynamics may involve hidden slow variables, for example, due to resonances. We propose an alignment algorithm for almost-periodic solution, in which case convergence of the parareal iterations is proved. The applicability of the method is demonstrated in numerical examples.
A theory of iterated averaging is developed for a class of highly oscillatory ordinary differential equations (ODEs) with three well separated time scales. The solutions of these equations are assumed to be (almost) periodic in the fastest time scales. It is proved that the dynamics on the slowest time scale can be approximated by an effective ODE obtained by averaging out oscillations. In particular, the effective dynamics of the considered class of ODEs is always deterministic and does not show any stochastic effects. This is in contrast to systems in which the dynamics on the fastest time scale is mixing. The systems are studied from three perspectives: first, using the tools of averaging theory; second, by formal asymptotic expansions; and third, by averaging with respect to fast oscillations using nested convolutions with averaging kernels. The latter motivates a hierarchical numerical algorithm consisting of nested integrators.
This article focuses on the mathematical problem of reconstructing dynamic permeability K(ω) of two-phase composites from data at different frequencies, utilizing the analytic structure of K(ω). To numerically simulate the data by solving the unsteady Stokes equations in the 3D domain reconstructed from µ-CT scans of the composites, the issue of segmentation is also addressed in this paper and an efficient segmentation scheme is suggested and implemented for cancellous bone.
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