Evaluation of Galerkin and Petrov-Galerkin model reduction for finite element approximations of the shallow water equations

- Galerkin and Petrov-Galerkin ROM introduced for semi-implicit FE schemes for SWE.
- Sufficient conditions for consistent snapshot collection methods are established.
- Numerical tests for Burgers equation and SWE confirm predictions for consistency.
- Both Galerkin and Petrov-Galerkin model reduction methods show comparable accuracy.

The shallow water equations (SWE) are widely used to model depth-averaged free-surface flows. They are well-studied, and many successful numerical methods have been posited for their solution. Nevertheless, accurate resolution of shallow-water flows can still be very computationally demanding. In particular, computational expense is a significant hurdle for inclusion of well-resolved shallow water models in engineering applications involving risk assessment, optimal design, or parameter estimation.The SWE are then a natural candidate for model reduction. In previous work by Lozovskiy et al. (2016), we presented an approach based on Proper Orthogonal Decomposition (POD) and hyper-reduction for a class of semi-implicit, stabilized finite element approximations of the SWE. Here, we revisit this scheme to further evaluate its consistency and performance using different snapshot collection procedures. We also consider a closely related Petrov-Galerkin technique based on least-squares minimization as an alternative to standard Galerkin projection. We formulate both schemes in a stabilized finite element context and examine their consistency rigorously both with and without hyper-reduction. We then evaluate their accuracy and robustness for test problems in one and two space dimensions.