In this paper, we investigate projection-based intrusive and data-driven non-intrusive model order reduction (MOR) methods in the numerical simulation of the rotating thermal shallow water equation (RTSWE) in parametric and non-parametric form. The RTSWE is a non-canonical Hamiltonian partial differential equation (PDE) with the associated conserved quantities, i.e., Hamiltonian (energy), mass, buoyancy, and the total vorticity. Discretization of the RTSWE in space with centered finite differences leads to Hamiltonian system of ordinary differential equations (ODEs) with linear and quadratic terms. The intrusive reduced-order model (ROM) is constructed with the proper orthogonal decomposition (POD) with the Galerkin projection (POD-G)  . We apply the operator inference (OpInf) with re-projection for non-intrusive reduced-order modeling. The OpInf non-intrusively learns a reduced model from state snapshot and time derivative which approximate the evolution of the RTSWE within a low-dimensional subspace. The OpInf data sampling scheme to obtain re-projected trajectories of the RTSWE corresponding to Markovian dynamics in low-dimensional subspaces, recovers of reduced models with high accuracy. In the parametric case, for both methods, we make use of the parameter dependency at the level of the PDE without interpolating between the reduced operators. The least-squares problem of the OpInf is regularized with the minimum norm solution. Both reduced models are able to accurately re-predict the training data and capture much of the overall system behavior in the prediction period. Due to re-projection, the OpInf is more costly than the POG-G, nevertheless, speed-up factors of order two are achieved for both ROMs. Numerical results demonstrate that the conserved quantities of the RTSWE are preserved for both ROMs over time and the long-term stability of the reduced solutions is achieved .
In this paper, we investigate projection-based intrusive and data-driven non-intrusive model order reduction methods in numerical simulation of rotating thermal shallow water equation (RTSWE) in parametric and non-parametric form. Discretization of the RTSWE in space with centered finite differences leads to Hamiltonian system of ordinary differential equations with linear and quadratic terms. The full-order model (FOM) is obtained by applying linearly implicit Kahan's method in time. Applying proper orthogonal decomposition with Galerkin projection (POD-G) , we construct the intrusive reduced-order model (ROM) . We apply operator inference (OpInf) with re-projection for non-intrusive reduced-order modeling. In the parametric case, we make use of the parameter dependency at the level of the PDE without interpolating between the reduced operators. The least-squares problem of the OpInf is regularized with the minimum norm solution. Both ROMs behave similar and are able to accurately predict the test and training data and capture system behavior in the prediction phase with several orders of computational speedup over the FOM. The preservation of system physics such as the conserved quantities of the RTSWE by both ROMs enables that the models fit better to data and stable solutions are obtained in long-term predictions, which are robust to parameter changes .
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