用于热力学一致系统的广义斜梯度嵌入
Generalized skew-gradient embedding for thermodynamically consistent systems
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中文总结 AI 辅助
研究针对热力学一致系统，提出广义斜梯度嵌入（GSGE）方法，通过将零能量贡献嵌入反对称算子实现。利用最小二乘法等确定规范，给出正则化近似等。将雅可比准则用于相关方程离散化，在多系统中展现良好性质，如保持结构、满足能量定律等。
中文摘要 AI 辅助
斜梯度嵌入（SGE）框架通过将零能量贡献嵌入反对称算子，将热力学一致系统重新表述为广义梯度流。在时间离散格式中，定义该算子的轮廓可在前一时间层评估，所得算子仍为反对称，其对离散能量平衡的贡献消失，这种显式处理常使多物理系统解耦。研究表明该算子不唯一，允许的规范形成仿射空间，称为广义斜梯度嵌入（GSGE）。对于正定度量，最小二乘法选择唯一的最小希尔伯特 - 施密特规范，自然度量可恢复SGE。此构造还给出正则化近似、非中性残差校正及保持规定不变量的规范。对于二阶规范，使用了充要的雅可比准则。将该准则应用于不可压缩纳维 - 斯托克斯方程的兼容MAC离散化，在半离散水平给出有限维二阶泊松 - GENERIC公式，隐式中点法则在全离散水平保持此二阶GENERIC结构并满足精确离散能量定律。对于Cahn - Hilliard - Navier - Stokes系统，正则化的GSGE - BDF2格式保持质量，无条件耗散离散能量，并允许解耦实现。
英文摘要
The skew-gradient embedding (SGE) framework~\cite{GuWangSGE2025} reformulates a thermodynamically consistent system as a generalized gradient flow by embedding its zero-energy contribution in a skew-symmetric operator. In a time-discrete scheme, the profiles defining this operator may be evaluated at previous time levels. The resulting operator remains skew-symmetric, so its contribution to the discrete energy balance vanishes; this explicit treatment often decouples multiphysics systems. We show that this operator is not unique: the admissible gauges form an affine space, and we call the resulting family generalized skew-gradient embeddings (GSGE). For any positive definite metric, least squares selects a unique minimum-Hilbert--Schmidt gauge, and the native metric recovers SGE. This construction also gives regularized approximations, corrections of non-neutral residuals, and gauges that preserve prescribed invariants. For rank-two gauges, we use a necessary and sufficient Jacobi criterion. Applying this criterion to a compatible MAC discretization of the incompressible Navier--Stokes equations gives 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 and satisfies 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 admits a decoupled implementation.
