讲座内容：

Preserving invariant regions (e.g., positivity of density and pressure, and the subluminal-velocity constraint in relativity) is fundamental to the physical consistency and mathematical well-posedness of hyperbolic conservation laws. However, developing high-order schemes that strictly respect nonlinear constraints remains a significant challenge for hyperbolic systems and computational fluid dynamics.In the first part of this talk, we introduce the Geometric Quasi-Linearization (GQL) framework. Building on key insights from convex geometry, GQL transforms arbitrary nonlinear convex constraints into  equivalent linear constraints by introducing free auxiliary variables. Crucially, GQL reveals a geometric insight: nonlinear constraints in the physical space are, in essence, linear when lifted to a higher-dimensional space. We establish the theoretical foundation of this framework and propose three effective construction methods, providing a unified approach to nonlinear invariant-domain-preserving analysis and design. In the second part, we apply GQL to the structure-preserving analysis of (ideal/relativistic) compressible magnetohydrodynamics (MHD), unveiling an intrinsic algebraic–differential coupling between thermodynamic constraints (pressure positivity) and the involution constraint (divergence-free magnetic field) at both the numerical and PDE levels. We prove that the preservation of algebraic bounds is generally conditioned on the discrete divergence-free structure. By deriving the exact compatibility condition, we construct high-order schemes that achieve synergistic preservation of both algebraic and differential constraints. The robustness of these methods is demonstrated through extreme simulations, including low plasma beta ($\approx 10^{-10}$) blast waves and astrophysical jets with Mach numbers up to $10^{6}$.