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 [4, 16, 18].
These methods - under the name of Hi-Mod approximation - 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 com-
promising 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.