Structure Preserving Model Order Reduction of Shallow Water Equations
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The two-dimensional rotational shallow water equations (SWEs) in the non-canonical Hamiltonian/Poisson form are integrated in time by the fully implicit average vector field (AVF) method, and in the f-plane as a partial differential equation (PDE) with quadratic nonlinearity by the linearly implicit Kahan's method. Reduced order models (ROMs) with proper orthogonal decomposition/discrete empirical interpolation method (POD/DEIM) preserve the Hamiltonian structure, and the tensorial POD the linear-quadratic structure of the SWE in the f-plane. We show that both methods preserve numerically the invariants like energy, the Casimirs like the enstrophy, mass, and circulation over a long time. The accuracy and computational efficiency of the ROMs are shown in numerical test problem.
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