Abstract
We introduce a new streamline derivative projection-based closure modeling strategy for the numerical stabilization of Proper Orthogonal Decomposition-Reduced Order Models (POD-ROM). As a first preliminary step, the proposed model is analyzed and tested for advection-dominated advection-diffusion-reaction equations. In this framework, the numerical analysis for the Finite Element (FE) discretization of the proposed new POD-ROM is presented, by mainly deriving the corresponding error estimates. Numerical tests for advection-dominated regime show the effciency of the proposed method, as well the increased accuracy over the standard POD-ROM that discovers its well-known limitations very soon in the numerical settings considered, i.e. for low diffusion coeffcients.
Highlights
Among the most popular Reduced Order Models (ROM) approaches, Proper Orthogonal Decomposition (POD) strategy provides optimal modes to represent the dynamics from a given database obtained by a full-order system
We have proposed a new stabilized POD-ROM for the numerical simulation of advectiondominated advection-diffusion-reaction equations
This model, denoted SD-POD-ROM, is derived from highorder stabilized Finite Element (FE) methods, and uses a streamline derivative projection-based operator to properly take into account the high frequencies advective derivative component of POD modes not included in the ROM
Summary
Among the most popular Reduced Order Models (ROM) approaches, Proper Orthogonal Decomposition (POD) strategy provides optimal (from the energetic point of view) modes to represent the dynamics from a given database (snapshots) obtained by a full-order system Onto these POD modes, a Galerkin projection of the governing equations can be employed to obtain a low-order dynamical system for the modes coefficients. We consider a Streamline Derivativebased (SD-based) approach used by Knobloch and Lube in [28] in the FE context, which only acts on the high frequencies of the advective derivative (see [1] for its extension to Navier–Stokes Equations (NSE)) This approach consists in adding a filtered advection stabilization term by basically following the streamlines to prevent spurious instabilities due do dominant advection.
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