Energy preserving reduced-order modeling of the rotating thermal shallow water equation

Abstract

In this paper, reduced-order models (ROMs) are developed for the rotating thermal shallow water equation (RTSWE) in the non-canonical Hamiltonian form with state-dependent Poisson matrix. The high fidelity full solutions are obtained by discretizing the RTSWE in space with skew-symmetric finite-differences, while preserving the Hamiltonian structure. The resulting skew-gradient system is integrated in time by the energy preserving average vector field (AVF) method. The ROM is constructed by applying proper orthogonal decomposition with the Galerkin projection, preserving the reduced skew-gradient structure, and integrating in time with the AVF method. The nonlinear terms of the Poisson matrix and Hamiltonian are approximated with the discrete empirical interpolation method to reduce the computational cost. The solutions of the resulting linear-quadratic reduced system are accelerated by the use of tensor techniques. 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 full-order model. Furthermore, we show that the ROMs are able to accurately predict the test and training data and capture the system behavior in the prediction phase.