A framework for optimal temporal reduced order modeling of nonlinear dynamical systems

An optimal temporal reduced order modeling framework is proposed for nonlinear dynamical systems. The governing equations are modified for an under-resolved simulation with an arbitrary scheme and a coarse temporal grid. Subgrid-scale models are developed to account for the unresolved temporal structure via inclusion of statistical information on a multi-point temporal stencil. These models are based upon principles of mean-square error minimization, conditional expectations and stochastic estimation. In order to validate the proposed framework, we investigate time-periodic solutions for a canonical Duffing oscillator using a coarse harmonic balance scheme. In order to demonstrate application of the proposed framework to a high dimensional nonlinear dynamical system, we also investigate a simply supported, geometrically nonlinear beam under the influence of time-periodic external forcing. For both problems, the subgrid-scale models are shown to significantly improve the accuracy of coarse-grained solutions.