Consistency of the full and reduced order models for evolve‐filter‐relax regularization of convection‐dominated, marginally‐resolved flows

Abstract

Numerical stabilization is often used to eliminate (alleviate) the spurious oscillations generally produced by full order models (FOMs) in under‐resolved or marginally‐resolved simulations of convection‐dominated flows. In this article, we investigate the role of numerical stabilization in reduced order models (ROMs) of marginally‐resolved, convection‐dominated incompressible flows. Specifically, we investigate the FOM–ROM consistency, that is, whether the numerical stabilization is beneficial both at the FOM and the ROM level. As a numerical stabilization strategy, we focus on the evolve‐filter‐relax (EFR) regularization algorithm, which centers around spatial filtering. To investigate the FOM‐ROM consistency, we consider two ROM strategies: (i) the EFR‐noEFR, in which the EFR stabilization is used at the FOM level, but not at the ROM level; and (ii) the EFR‐EFR, in which the EFR stabilization is used both at the FOM and at the ROM level. We compare the EFR‐noEFR with the EFR‐EFR in the numerical simulation of a 2D incompressible flow past a circular cylinder in the convection‐dominated, marginally‐resolved regime. We also perform model reduction with respect to both time and Reynolds number. Our numerical investigation shows that the EFR‐EFR is more accurate than the EFR‐noEFR, which suggests that FOM‐ROM consistency is beneficial in convection‐dominated, marginally‐resolved flows.

A novel Large Eddy Simulation model for the Quasi-Geostrophic Equations in a Finite Volume setting

“…A sensitivity analysis for the filtering radius [2] would help us understand how to obtain the most accurate results when compared to a direct numerical simulation. Moreover, it would be interesting to test the performance of a class of deconvolution-based indicator functions and to implement an efficient algorithm called Evolve-Filter-Relax, which proved to work well for the Leray-α model [3,11,9,10,12,13,32]. Thus, we believe they could be successful also for the BV-NL-α model.…”

Residual-based stabilized reduced-order models of the transient convection-diffusion-reaction equation obtained through discrete and continuous projection

“…Model consistency is the setting in which the same stabilization method is used in the FOM and ROM. We note that, when the same parameters are used in the FOM and ROM (i.e., we have parameter FOM-ROM consistency [146]), model consistency is a special class of Type 2 consistency. In [111] (see also [52]), the authors have argued both numerically and theoretically (in particular, see Section 3.3 and Proposition 3.1 in [111]) that using the same type of stabilization (i.e., SUPG) in the FOM and ROM yields more accurate ROM results.…”

“…In [111] (see also [52]), the authors have argued both numerically and theoretically (in particular, see Section 3.3 and Proposition 3.1 in [111]) that using the same type of stabilization (i.e., SUPG) in the FOM and ROM yields more accurate ROM results. More recently, model consistency for the evolve-filter-relax ROM [146] (which is a spatial filtering-based stabilization, such as those described in Section 5) was shown to increase the ROM accuracy.…”

“…The ROM is then executed with the same stabilized form used to generate the snapshots; this is the so-called "offline-online stabilization strategy" outlined in Ref. [4] (see also [146]), which comprises a form of model consistency. This procedure, unfortunately, makes it difficult to compare the performance of different stabilization techniques; i.e., if we generated the FOM solution with SUPG, then comparing SUPG ROM solutions to LSPG ROM solutions becomes unfair.…”

“…While, in this paper, we focus on "residual-based" stabilization techniques, we mention that a large body of work has been dedicated to alternative approaches targeting this issue. This work includes, but is not limited to: stabilizing inner products that guarantee a non-increasing energy [83,82,13,128,138] or non-decreasing entropy [81,34]; stabilizing subspace rotations that account for truncated modes a priori [11,10]; eigenvalue reassignment methods that calculate a stabilizing correction to a given linear [84] or nonlinear [124] ROM that is found to be unstable after it is constructed; structure preserving methods that guarantee that the ROM satisfy physical constraints [30,93,32,15,35,47,57]; spatial filtering-based stabilization methods [58,73,85,146,158] that filter out unphysical high-frequency content, inf-sup stabilization methods that enforce the inf-sup condition in the incompressible Stokes and Navier-Stokes equations [17,26,45]; and closure modeling approaches [3,157,17,156,155,137] that add additional "closure" terms to the ROM so-as to account for the impact of truncated modes.…”