Overview
The research presents a generalized skew-gradient embedding (GSGE) framework, which builds upon the existing skew-gradient embedding (SGE) framework. The SGE framework reformulates thermodynamically consistent systems into a generalized gradient flow by incorporating a zero-energy contribution within a skew-symmetric operator. The GSGE framework explores the non-uniqueness of this operator, identifying an affine space of admissible gauges.
Research Context
The SGE framework, outlined in GuWangSGE2025, converts thermodynamically consistent systems into a generalized gradient flow. This conversion involves embedding the system's zero-energy contribution within a skew-symmetric operator. When implemented in a time-discrete scheme, the profiles used to define this operator can be evaluated at preceding time levels. The property of skew-symmetry is maintained, ensuring that the operator's contribution to the discrete energy balance is zero. This explicit handling can facilitate the decoupling of multiphysics systems.
Approach
The core of the GSGE approach is the recognition that the skew-symmetric operator within the SGE framework is not unique. The researchers identified that the admissible gauges for this operator form an affine space, leading to the designation of the resulting family as generalized skew-gradient embeddings (GSGE). For any positive definite metric, a unique minimum-Hilbert–Schmidt gauge can be selected using a least squares method. The native metric is shown to recover the original SGE formulation. The construction of GSGE allows for the generation of regularized approximations and corrections for non-neutral residuals. Furthermore, gauges can be designed to preserve specific invariants. For rank-two gauges, the research utilizes a necessary and sufficient Jacobi criterion.
Findings
- The skew-symmetric operator in the SGE framework is not unique; admissible gauges form an affine space, defining the GSGE framework.
- A unique minimum-Hilbert–Schmidt gauge can be selected via least squares for any positive definite metric, and the native metric recovers the SGE.
- The GSGE construction enables regularized approximations, corrections of non-neutral residuals, and gauges that preserve prescribed invariants.
- For rank-two gauges, a necessary and sufficient Jacobi criterion is employed.
- Applying this criterion to a compatible MAC discretization of the incompressible Navier–Stokes equations yielded a finite-dimensional rank-two Poisson–GENERIC formulation at the semi-discrete level.
- The implicit midpoint rule preserves this rank-two GENERIC structure at the fully discrete level for the incompressible Navier–Stokes equations, satisfying the exact discrete energy law.
- For the Cahn–Hilliard–Navier–Stokes system, the regularized GSGE–BDF2 scheme preserves mass, dissipates the discrete energy unconditionally, and supports a decoupled implementation.