  In this paper, Hamiltonian and energy preserving reduced-order models are
developed for the rotating thermal shallow water equation (RTSWE) in the
non-canonical Hamiltonian form with the state-dependent Poisson matrix. The
high fidelity full solutions are obtained by discretizing the RTSWE in space
with skew-symmetric finite-differences, that preserve the Hamiltonian
structure. The resulting skew-gradient system is integrated in time with the
energy preserving average vector field (AVF) method. The reduced-order model
(ROM) is constructed in the same way as the full order model (FOM), preserving
the reduced skew-symmetric structure and integrating in time with the AVF
method. Relying on structure-preserving discretizations in space and time and
applying proper orthogonal decomposition (POD) with the Galerkin projection, an
energy preserving reduced order model (ROM) is constructed. The nonlinearities
in the ROM are computed by applying the discrete empirical interpolation (DEIM)
method to reduce the computational cost. The computation of the reduced-order
solutions is accelerated further by the use of tensor techniques. The overall
procedure yields a clear separation of the offline and online computational
cost of the reduced solutions. The accuracy and computational efficiency of the
ROMs are demonstrated for a numerical test problem. Preservation of the energy
(Hamiltonian), and other conserved quantities, i.e. mass, buoyancy, and total
vorticity show that the reduced-order solutions ensure the long-term stability
of the solutions while exhibiting several orders of magnitude computational
speedup over the FOM.
