Time, space, and space-time hybrid clustering POD with application to the Burgers´ equation

In this paper, we investigate new methods that make the Proper Orthogonal Decomposition (POD) more accurate in reducing the order of large scale nonlinear systems. The general framework is to apply POD locally to clusters instead of applying it to the global system. Each cluster contains relatively close in distance behavior within itself, and considerably far with respect to other clusters. We introduce three different clustering schemes in time, space and space-time. For time clustering, time snapshots of the solution are grouped into clusters where the solution exhibits significantly different features and a local basis is pre-computed and assigned to each cluster. Space clustering is done in a similar fashion for the space vectors of the solution instead of snapshots, and finally space-time clustering is applied through a hybrid clustering scheme that combines space and time behavior together. We apply our method to reduce a nonlinear convective PDE system governed by the Burgers´ equation for fluid flows over 1D and 2D domains and show a significant improvement over conventional POD.