Hybrid reduced-order integration with proper orthogonal decomposition and dynamic mode decomposition

A data-driven hybrid numerical integrator is introduced to exploit, numerically, the formation of nonlinear coherent structures that often appear in nonlinear PDEs. Full simulations of the PDE allow model reduction algorithms such as the proper orthogonal decomposition and dynamic mode decomposition to generate reduced order models in an "online" manner. Criteria based on the comparison of these two independent reduction techniques, similar to model predictive control, determine whether the reduced model is accurate without direct evaluation of the underlying PDE. The method is implemented and explored for two prototypical PDE example models and significantly reduces the computational cost of solving those equations even when bifurcations occur. © 2013 Society for Industrial and Applied Mathematics.