A priori space-time separated representation for the reduced order modeling of low Reynolds number flows

Reduced order models (ROMs) in fluid dynamics are nowadays mostly developed by performing a projection of the Navier-Stokes equations onto a low-dimensional space basis. This basis is usually obtained through Proper Orthogonal Decomposition (POD), which remains one the most efficient techniques to compress precomputed data. The main drawback of a posteriori POD based ROMs is however their lacks of reliability as their parameters are varied, preventing their direct uses within optimization algorithms. The goal of the present article is to obtain an a priori low-dimensional space-time separated representation of the fluid fields, without precomputed data. The approach is based on the use of space-time Proper Generalized Decomposition (PGD) definitions, which are successfully applied in several fields but whose uses in fluid dynamics remain scarce. Their applications to the Navier-Stokes equations are indeed not straightforward, due to the pressure-velocity coupling, the divergence-free constraint and the non-linear convective term. The ROMs are built here from a space-time weak formulation of the Chorin-Temam prediction-correction scheme. More particularly, a priori space-time separated representations are obtained by applying the Galerkin based progressive PGD definition. The minimax PGD definition is moreover experimented on a linear Stokes simplified case. The related algorithms are explicitly given and illustrated on a transient lid-driven cavity flow. It is shown that the a priori space-time separated representations converge to the full model solution as the decomposition order increases. The ability of the resulting ROM to learn iteratively from its own error is highlighted: the progressive PGD algorithm can be used to effectively enrich incomplete POD and PGD precomputed space bases, such as those obtained with different parameter values. This may allow to ensure the accuracy of a ROM as the parameter is varied, which is of crucial interest for optimization problems.