Dimensionality reduction of convection-dominated flows on an optimally morphing grid

Foundations and preliminary results of a new projection-based model order reduction approach are summarized. The method is specifically designed for convection dominated nonlinear fluid flows. In this method, the evolution of the flow is approximated on an optimally morphing grid. The low-rank grid deformation, a solution of an optimization problem, is generated in such a way that the low-dimensional representation of the states on this morphing grid has lower error when compared to traditional POD on an Eulerian grid. Global basis functions are used to approximate the state variables on the low-rank grid. It is demonstrated that in this framework, certain wave-like solutions exhibit low-rank structure and thus, can be efficiently compressed using relatively few global bases. The proposed approach is successfully demonstrated for the reduction of several representative 1D and 2D problems, featuring nonlinearities, and bi-directional waves with different boundary conditions and is compared with the traditional method on the Eulerian grid.