# Structure Preserving Model Order Reduction of Shallow Water Equations

## TL;DR

This paper introduces two structure-preserving reduced-order modeling techniques for the 2D shallow water equations, ensuring long-term stability and conservation of invariants like energy and enstrophy.

## Contribution

It presents novel ROM construction methods that preserve Hamiltonian and quadratic structures of the SWE, enhancing stability and accuracy over long simulations.

## Key findings

- Both methods preserve invariants such as energy and enstrophy.
- The ROMs demonstrate high accuracy and computational efficiency.
- Long-term stability of the solutions is achieved.

## 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.