Efficient Non-Linear Proper Orthogonal Decomposition (POD)/Galerkin Reduced Order Models with Stable Penalty Enforcement of Boundary Conditions

An efficient, stability-preserving model reduction technique for non-linear initial boundary value problems (IBVPs) 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 penaltyenforced 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 (CDR) 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 (BPIM) proposed by Peraire, Nguyen et al. in [22, 23]. 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. Copyright c © 2011 John Wiley & Sons, Ltd.