Reduced order models based on Markovian and Non-Markovian frameworks

Empirical mode decomposition for reduced order modeling like POD have been used for efficient dimensionality reduction. However, strong mode truncation may lead to stabilization problems owing to the decoupling of linear and non-linear part of the solutions. Hence, stabilization techniques inspired by turbulence closure model were proposed in the past. Although this process is often effective in stabilization but cannot prevent the information loss which can be essential sometimes for mixing efficiency. A multi-scale approach to model the microscopic dynamics which captures the macroscopic physics of the complex system is presented here. Two novel ideas are pursued 1) to build a simple one dimensional scale preserving reduced order model via the Kramers Moyal expansion 2) to reduce the dimensionality while preserving the smallest scales by projecting the fast decorrelating modes on to the slow modes via the Mori-Zwanzig projection operator. The developed reduced order model i.e. a generalized Langevin equation solves the mode decoupling problem introduced by POD. The model so developed can be used both as a standalone microscopic model of the system and for turbulence closure. Results are presented for a classical test problem and a mixed convective liquid metal channel flow.