Efficient non‐linear proper orthogonal decomposition/Galerkin reduced order models with stable penalty enforcement of boundary conditions

Abstract

An efficient, stability-preserving model reduction technique for non-linear initial boundary value problems whose solutions exhibit inherently non-linear dynamics such as metastability and periodic regimes (limit cycles) is developed. The approach is based on the 'continuous' Galerkin projection approach in which the continuous governing equations are projected onto the reduced basis modes in a continuous inner product. The reduced order model (ROM) basis is constructed via a proper orthogonal decomposition (POD). In general, POD basis modes will not satisfy the boundary conditions of the problem. A weak implementation of the boundary conditions in the ROM based on the penalty method is developed. Asymptotic stability of the ROM with penalty-enforced boundary conditions is examined using the energy method, following linearization and localization of the governing equations in the vicinity of a stable steady solution. This analysis, enabled by the fact that a continuous representation of the reduced basis is employed, leads to a model reduction method with an a priori stability guarantee. The approach is applied to two non-linear problems: the Allen-Cahn (or 'bistable') equation and a convection-diffusion-reaction system representing a tubular reactor. For each of these problems, bounds on the penalty parameters that ensure asymptotic stability of the ROM solutions are derived. The non-linear terms in the equations are handled efficiently using the 'best points' interpolation method proposed recently by Peraire, Nguyen et al. Numerical experiments reveal that the POD/Galerkin ROMs with stability-preserving penalty boundary treatment for the two problems considered, both without as well as with interpolation, remain stable in a way that is consistent with the solutions to the governing continuous equations and capture the correct non-linear dynamics exhibited by the exact solutions to these problems. Published 2012.

POD‐identification reduced order model of linear transport equations for control purposes

“…The range of the factor for which the solution is stable can for instance be determined using Poincaré maps . In addition, the bounds on the factor that ensure asymptotic stability of the ROM can be derived . The prediction error increases when the frequency of the time‐dependent BC is doubled.…”

“…29 In addition, the bounds on the factor that ensure asymptotic stability of the ROM can be derived. 28 The prediction error increases when the frequency of the time-dependent BC is doubled. There are two ways to increase the number of snapshots in order to reduce the error and to enhance stability.…”

“…There is no problem in the case of homogeneous BCs as the linear combinations of snapshots with homogeneous boundary conditions will naturally satisfy the same BCs. Another way to deal with nonhomogeneous BCs is to add an additional constraint to the transport equation in order to weakly enforce the boundary condition for the ROM with a penalty factor . No modification of the snapshots is needed other than adding the boundary points in case these are not present; otherwise, the ROM could become unstable.…”

“…Another way to deal with nonhomogeneous BCs is to add an additional constraint to the transport equation in order to weakly enforce the boundary condition for the ROM with a penalty factor. 3,23,25,28,29 No modification of the snapshots is needed other than adding the boundary points in case these are not present; otherwise, the ROM could become unstable. The constraint is added to the transport equation in the following way:…”

A hybrid stabilization approach for reduced‐order models of compressible flows with shock‐vortex interaction

“…ROMs developed by POD-Galerkin projection suffer from different instabilities in many cases. [12][13][14] Barone et al 4 pointed out that ROM instability may come from the lack of an energy-conserving definition for the inner product used in both the POD computation and the Galerkin projection, and they suggested to use a symmetry inner product which leads to a ROM in the form of a symmetric operator being applied to primary variables. The symmetry inner product has shown promising results in other applications.…”