Hi-POD solution of parametrized fluid dynamics problems: Preliminary results

Numerical modeling of fluids in pipes or network of pipes (like in the circulatory system) has been recently faced with new methods that exploit the specific nature of the dynamics, so that a one dimensional axial mainstream is enriched by local secondary transverse components (Ern et al., Numerical Mathematics and Advanced Applications, pp 703-710. Springer, Heidelberg, 2008; Perotto et al., Multiscale Model Simul 8(4):1102-1127, 2010; Perotto and Veneziani, J Sci Comput 60(3):505-536, 2014). These methods—under the name of Hierarchical Model (Hi-Mod) reduction—construct a solution as a finite element axial discretization, completed by a spectral approximation of the transverse dynamics. It has been demonstrated that Hi-Mod reduction significantly accelerates the computations without compromising the accuracy. In view of variational data assimilation procedures (or, more in general, control problems), it is crucial to have efficient model reduction techniques to rapidly solve, for instance, a parametrized problem for several choices of the parameters of interest. In this work, we present some preliminary results merging Hi-Mod techniques with a classical Proper Orthogonal Decomposition (POD) strategy. We name this new approach as Hi-POD model reduction. We demonstrate the efficiency and the reliability of Hi-POD on multiparameter advection-diffusion-reaction problems as well as on the incompressible Navier-Stokes equations, both in a steady and in an unsteady setting.