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arXiv · 1907.09406

Structure Preserving Model Order Reduction of Shallow Water Equations

Bülent Karasözen·Süleyman Yıldız·Murat Uzunca
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Abstract

In this paper, we present two different approaches for constructing reduced-order models (ROMs) for the two-dimensional shallow water equation (SWE). The first one is based on the noncanonical Hamiltonian/Poisson form of the SWE. After integration in time by the fully implicit average vector field method, ROMs are constructed with proper orthogonal decomposition/discrete empirical interpolation method (POD/DEIM) that preserves the Hamiltonian structure. In the second approach, the SWE as a partial differential equation with quadratic nonlinearity is integrated in time by the linearly implicit Kahan's method and ROMs are constructed with the tensorial POD that preserves the linear-quadratic structure of the SWE. We show that in both approaches, the invariants of the SWE such as the energy, enstrophy, mass, and circulation are preserved over a long period of time, leading to stable solutions. We conclude by demonstrating the accuracy and the computational efficiency of the reduced solutions by a numerical test problem.

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Bülent Karasözen, Süleyman Yıldız, Murat Uzunca. 2019-07-10. Structure Preserving Model Order Reduction of Shallow Water Equations. https://doi.org/10.1002/mma.6751

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