  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.
