Reduced-order modelling of shallow water equations

Yıldız, Süleyman
The shallow water equations (SWEs) consist of a set of two-dimensional partial differential equations (PDEs) describing a thin inviscid fluid layer flowing over the topography in a frame rotating about an arbitrary axis. SWEs are widely used in modeling large-scale atmosphere/ocean dynamics and numerical weather prediction. High-resolution simulations of the SWEs require long time horizons over global scales when combined with accurate resolution in time and space makes simulations very time-consuming. While high-resolution ocean-modeling simulations are still feasible on large HPC machines, performing many query applications, such as repeated evaluations of the model over a range of parameter values, at these resolutions, is not feasible. Reduced-order modeling enables fast simulation of the PDEs using high-fidelity solutions. In this thesis, reduced-order models (ROMs) are investigated for the rotating SWE, with constant (RSWE) and non-traditional SWE with full Coriolis force (NTSWE), and for rotating thermal SWE (RTSWE) while preserving their non-canonical Hamiltonian-structure, the energy, and Casimir’s, i.e. mass, enstrophy, vorticity, and buoyancy. Two different approaches are followed for constructing ROMs; the traditional intrusive model order reduction with Galerkin projection and the data-driven, non-intrusive ROMs. The full order models (FOM) of the SWE, which needed to construct the ROMs are obtained by discretizing the SWE in space by finite differences by preserving the skew-symmetric structure of the Poisson matrix. Applying intrusive proper orthogonal decomposition (POD) with the Galerkin projection, energy preserving ROMs are constructed for the NTSWE and RTSWE in skew gradient form. Due to the nonlinear terms, the dimension of the reduced-order system scales with the dimension of the FOM. The nonlinearities in the ROM are computed by applying the discrete empirical interpolation (DEIM) method to reduce the computational cost. The computation of the reduced-order solutions is accelerated further by the use of tensor techniques. For the RSWE in linear-quadratic form, the dimension of the reduced solutions is obtained using tensor algebra without necessitating hyper-reduction techniques like the DEIM. Applying POD in a tensorial framework by exploiting matricizations of tensors, the computational cost is further reduced for the rotating SWE in linear-quadratic as well in skew-gradient form. In the data-driven, non-intrusive ROMs are learned only from the snapshots by solving an appropriate least-squares optimization problem in a low-dimensional subspace. Data-driven ROMs are constructed for the NTSWE and RTSWE with the operator inference (OpInf). Computational challenges such as ill-conditioning and regularization are discussed. The non-intrusive model order reduction framework is extended to a parametric case, whereas we make use of the parameter dependency at the level of the PDE without interpolating between the reduced operators. The intrusive and non-intrusive ROMs in linear-quadratic and skew-gradient form yield a clear separation of the offline and online computational costs. Both ROM behave similarly and can accurately predict in the test and training data and capture system behavior in the prediction phase. The preservation of physical quantities in the ROMs of the SWEs such as energy (Hamiltonian), and other conserved quantities, i.e., mass, buoyancy, and total vorticity, enables that the models fit better to data and stable solutions are obtained in the long-term predictions which are robust to parameter changes while exhibiting several orders of magnitude computational speed-up over the FOMs.


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Citation Formats
S. Yıldız, “Reduced-order modelling of shallow water equations,” Ph.D. - Doctoral Program, Middle East Technical University, 2021.