arXiv subjects

Yoonsang Lee

Publications and source records attributed to Yoonsang Lee.

39 records · Page 3Linked to original sources

arXiv2015

Fast integrators for dynamical systems with several temporal scales

We propose a fast integrator to a class of dynamical systems with several temporal scales. The proposed method is developed as an extension of the variable step size Heterogeneous Multiscale Method (VSHMM), which is a two-scale integrator developed by the authors. While iterated applications of multiscale integrators for two different scales increase the computational complexity exponentially as the number of different scales increases, the proposed method, on the other hand, has computational complexity linearly proportional to the number of different scales. This efficiency is achieved by solving different scale components of the vector fields with variable time steps. It is shown that variable time stepping of different force components has an effect of fast integration for the effective force of the slow dynamics. The proposed fast integrator is numerically tested on problems with several different scales which are dissipative and highly oscillatory including multiscale partial differential equations with sparsity in the solution space.

math.NA↗

arXiv2015

Multiscale numerical methods for passive advection-diffusion in incompressible turbulent flow fields

We propose a seamless multiscale method which approximates the macroscopic behavior of the passive advection-diffusion equations with steady incompressible velocity fields with multi-spatial scales. The method uses decompositions of the velocity fields in the Fourier space, which are similar to the decomposition in large eddy simulations. It also uses a hierarchy of local domains with different resolutions as in multigrid methods. The effective diffusivity from finer scale is used for the next coarser level computation and this process is repeated up to the coarsest scale of interest. The grids are only in local domains whose sizes decrease depending on the resolution level so that the overall computational complexity increases linearly as the number of different resolution grids increases. The method captures interactions between finer and coarser scales but has to sacrifice some of the interaction between the fine scales. The proposed method is numerically tested with 2D examples including a successful approximation to a continuous spectrum flow.

math.NA↗

arXiv2013

Variable Step Size Multiscale Methods for Stiff and Highly Oscillatory Dynamical Systems

We present a new numerical multiscale integrator for stiff and highly oscillatory dynamical systems. The new algorithm can be seen as an improved version of the seamless Heterogeneous Multiscale Method by E, Ren, and Vanden-Eijnden and the method FLAVORS by Tao, Owhadi, and Marsden. It approximates slowly changing quantities in the solution with higher accuracy than these other methods while maintaining the same computational complexity. To achieve higher accuracy, it uses variable mesoscopic time steps which are determined by a special function satisfying moment and regularity conditions. Detailed analytical and numerical comparison between the different methods are given.

math.NA↗