发表机构
School of Science, Sun Yat-Sen University; Institute of Mathematics, University of Zürich; Department of Mathematics and Shenzhen International Center for Mathematics, Southern University of Science and Technology（中山大学理学院; 苏黎世大学数学研究所; 南方科技大学数学系及深圳国际数学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出统一不变域保持框架，通过障碍-勒让德变换和单元平均分解等机制，在一般网格上实现双曲方程混合离散格式的严格可容许性保持，并开发了三阶格式。
AI 中文摘要
本文针对双曲守恒律的混合离散格式，包括主动通量和PAMPA方法，提出了一个统一的不变域保持（IDP）框架。在该框架中，单元平均值通过守恒方式更新，而单元边界点值则在可能非守恒的算子下演化。主要挑战在于，在单一局部稳定性条件下，保持这两组耦合状态的可容许性，而无需增加演化自由度或依赖更新后的修复。对于点值更新，我们引入了一种基于障碍-勒让德映射的可容许性变换，该变换适用于由凹约束描述的凸可容许内部，并证明了逆映射在相关集合上全局定义且利普希茨连续。对于守恒的单元平均值更新，我们建立了一个结构障碍定理：仅由可容许边界迹构造的单状态连续物理通量，当内部重构值不受控制时，对于任何给定的CFL数，都无法提供守恒的IDP保证。这识别出必须由额外IDP通量机制提供的缺失局部控制。为了显式实现该机制，我们结合单元平均分解（CAD）、几何拟线性化和局部先验缩放，在通量评估前构造可容许的、通常不连续的迹状态。在基于迹的显式CFL条件下，常数由局部CAD权重和迹状态波速界限确定，耦合混合更新保持预定的不变域。在三角形、笛卡尔、凸四边形和一般凸多边形网格上开发了具体的三阶格式。数值结果展示了光滑解的设计精度阶数和物理可容许性的严格保持。
英文摘要
This paper presents a unified invariant-domain-preserving (IDP) framework for hybrid discretizations of hyperbolic conservation laws, including active flux and PAMPA methods, in which cell averages are updated conservatively while cell-boundary point values evolve under a possibly non-conservative operator. The main challenge is to preserve admissibility for these two coupled sets of states under a single local stability condition, without adding evolved degrees of freedom or relying on post-update repairs. For the point-value update, we introduce an admissibility transform based on a barrier--Legendre map for convex admissible interiors described by concave constraints, and prove that the inverse map is globally defined and Lipschitz continuous on the relevant sets. For the conservative cell-average update, we establish a structural obstruction theorem: the single-state continuous physical flux built from admissible boundary traces alone cannot provide a conservative IDP guarantee, for any prescribed CFL number, when internal reconstruction values are uncontrolled. This identifies the missing local control that must be supplied by an additional IDP flux mechanism. To realize this mechanism explicitly, we combine cell average decompositions (CAD), geometric quasilinearization, and local a priori scaling to construct admissible, generally discontinuous trace states before flux evaluation. Under an explicit trace-based CFL condition, with constants determined by local CAD weights and trace-state wave-speed bounds, the coupled hybrid update preserves the prescribed invariant domain. Concrete third-order schemes are developed on triangular, Cartesian, convex quadrilateral, and general convex polygonal meshes. Numerical results demonstrate the designed order of accuracy for smooth solutions and the strict preservation of physical admissibility.
