  The metriplectic formalism is useful for describing complete dynamical
systems which conserve energy and produce entropy. This creates challenges for
model reduction, as the elimination of high-frequency information will
generally not preserve the metriplectic structure which governs long-term
stability of the system. Based on proper orthogonal decomposition, a provably
convergent metriplectic reduced-order model is formulated which is guaranteed
to maintain the algebraic structure necessary for energy conservation and
entropy formation. Numerical results on benchmark problems show that the
proposed method is remarkably stable, leading to improved accuracy over long
time scales at a moderate increase in cost over naive methods.
