Energy preserving reduced-order modelling of thermal shallow water equation

TL;DR

This work develops energy- and structure-preserving reduced-order models for the rotating thermal shallow water equation (RTSWE) in a non-canonical Hamiltonian framework with a state-dependent Poisson matrix. By combining space-time structure-preserving discretizations, POD-Galerkin projection, DEIM hyper-reduction, and tensor techniques, the authors construct a reduced-order model that preserves the Hamiltonian (energy) and Casimirs, ensuring long-time stability with significant online speedups. The reduced system uses separate POD bases for each state variable, along with DEIM for nonlinear terms and Q-DEIM for selection, enabling efficient online computation. Numerical experiments on a double-vortex test demonstrate accurate, energy- and invariant-preserving ROMs with substantial speedups compared to the full-order model, highlighting the approach’s potential for long-time geophysical fluid simulations.

Abstract

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.

Figures (4)

• Figure 1: Normalized singular values: (left) state variables, (middle and right) nonlinear terms
• Figure 2: Bouyancy at the final time
• Figure 3: Potential vorticity at the final time
• Figure 4: Relative errors in the conserved quantities.