Skew Gradient Embedding for Thermodynamically Consistent Systems

Abstract: We propose a novel Skew Gradient Embedding (SGE) framework for systematically reformulating thermodynamically consistent partial differential equation (PDE) models-capturing both reversible and irreversible processes-as generalized gradient flows. These models include a wide spectrum of models in classical electrodynamics, fluid mechanics, quantum mechanics, rheology of complex fluids, solid mechanics, and statistical physics. Exploiting the intrinsic structure of generalized gradient flow models, especially, the skew symmetric component expressed by the exterior 2-form, we develop a unified stabilization strategy for constructing numerical schemes that either preserve the energy dissipation rate or ensure discrete energy stability. This stabilization strategy enables the design of both first- and second-order schemes, highlighting the flexibility and generality of the SGE approach in algorithm development. A key strength of SGE is its flexible treatment of skew-gradient (zero-energy-contribution) terms arising from reversible dynamics either implicitly or explicitly. While treated explicitly, it often leads to a natural decoupling of the governing equations in multiphysics systems, thereby improving computational efficiency without compromising stability or accuracy. Numerical experiments confirm the robustness, accuracy, and performance advantages of the proposed schemes.