General numerical framework to derive structure preserving reduced order models for thermodynamically consistent reversible-irreversible PDEs

In this paper, we propose a general numerical framework to derive structure-preserving reduced-order models for thermodynamically consistent PDEs. Our numerical framework has two primary features: (a) a systematic way to extract reduced-order models for thermodynamically consistent PDE systems while maintaining their inherent thermodynamic principles and (b) a general process to derive accurate, efficient, and structure-preserving numerical algorithms to solve these reduced-order models. The platform's generality extends to various PDE systems governed by embedded thermodynamic laws, offering a unique approach from several perspectives. First, it utilizes the generalized Onsager principle to transform the thermodynamically consistent PDE system into an equivalent form, where the free energy of the transformed system takes a quadratic form in terms of the state variables. This transformation is known as energy quadratization (EQ). Through EQ, we gain a novel perspective on deriving reduced-order models that continue to respect the energy dissipation law. Secondly, our proposed numerical approach automatically provides algorithms to discretize these reduced-order models. The proposed algorithms are always linear, easy to implement and solve, and uniquely solvable. Furthermore, these algorithms inherently ensure the thermodynamic laws. Our platform offers a distinctive approach for deriving structure-preserving reduced-order models for a wide range of PDE systems with underlying thermodynamic principles.