Energy preserving reduced-order models are developed for the rotating thermal shallow water (RTSW) equation in the non-canonical Hamiltonian /Poisson form . The RTSW equation is discretized in space by the skew-symmetric finite-difference operators to preserve the Hamiltonian structure. The resulting system of ordinary differential equations is integrated in time by the energy preserving average vector field (AVF) method. An energy preserving, computationally efficient reduced-order model (ROM) is constructed by proper orthogonal decomposition (POD) with the Galerkin projection  . The nonlinearities in the ROM are efficiently computed by discrete empirical interpolation method (DEIM) . Preservation of the energy (Hamiltonian), and other conserved quantities ; total mass, total buoyancyand total potential vorticity , by the reduced-order solutions are demonstrated which ensures the long term stability of the reduced-order solutions . The accuracy and computational efficiency of the ROMs are shown by a numerical test problem .
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 .
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Energy
In this paper, Hamiltonian and energy
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(RTSW) equation
equation (RTSWE)
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/Poisson form . The RTSW equation is discretized in space by the
form with the state-dependent Poisson matrix. The high fidelity full solutions are obtained by discretizing the RTSWE in space with
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finite-difference operators to
finite-differences, that
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system of ordinary differential equations
skew-gradient system
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by
with
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An energy preserving, computationally efficient
The
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by
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
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, an energy preserving reduced order model (ROM) is constructed
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efficiently computed by
computed by applying the
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method
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.
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.
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; total mass, total buoyancyand total potential vorticity , by
, i.e. mass, buoyancy, and total vorticity show that
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are demonstrated which ensures the long term
ensure the long-term
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reduced-order solutions . The accuracy and computational efficiency of the ROMs are shown by a numerical test problem
solutions while exhibiting several orders of magnitude computational speedup over the FOM