Structure-preserving reduced order modelling of thermal shallow water equation
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Energy preserving reduced-order models are developed for the rotating thermal shallow water (RTSW) equation in the non-canonical Hamiltonian/Poisson form. The RTSW equation is discretized in space by the skew-symmetric finite-difference operators to preserve the Hamiltonian structure. The resulting system of ordinary differential equations is integrated in time by the energy preserving average vector field (AVF) method. An energy preserving, computationally efficient reduced-order model (ROM) is constructed by proper orthogonal decomposition (POD) with the Galerkin projection. The nonlinearities in the ROM are efficiently computed by discrete empirical interpolation method (DEIM). Preservation of the energy (Hamiltonian), and other conserved quantities; total mass, total buoyancy and total potential vorticity, by the reduced-order solutions are demonstrated which ensures the long term stability of the reduced-order solutions. The accuracy and computational efficiency of the ROMs are shown by a numerical test problem.
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