信息几何正则化可压缩欧拉方程的间断伽辽金半离散化
Discontinuous Galerkin Semidiscretization of the Information Geometric Regularized Compressible Euler Equations
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中文总结 AI 辅助
本研究开发了信息几何正则化框架的间断伽辽金离散化方法，无需激波捕捉限制器即可稳定可压缩欧拉流的激波，且随多项式阶数提升能解析更精细流特征，同时保留解的光滑区域特性。
中文摘要 AI 辅助
可压缩欧拉流中的激波稳定化仍是高阶数值方法面临的核心挑战。现有激波捕捉方法包括限制器、人工黏性及基于重构的方法，在鲁棒性、精度、细尺度流特征保留与计算复杂度之间存在权衡。本研究针对Cao和Schäfer提出的信息几何正则化（IGR）框架，开发了可压缩欧拉方程的间断伽辽金（DG）离散化方法。该方法在偏微分方程（PDE）层面通过向欧拉方程添加从辅助椭圆方程得到的熵压Σ来正则化激波。在DG格式中，正则化项仅通过增广压力P+Σ进入欧拉通量，在双曲型与椭圆型方程使用共同近似空间的同时，保留了离散化的守恒结构。涵盖一维和二维基准问题的数值实验表明，所提格式无需激波捕捉限制器或人工黏性即可稳定激波，尽管当密度或压力趋近于零时可能仍需保正方法。与特征TVB限制DG格式相比，IGR-DG方法随多项式阶数提升可解析更精细的流特征，同时保持稳定的激波解析。熵压仅定位于强压缩区域，在流场光滑区域激活极少，在保留其余区域解的同时提供了选择性的PDE层面正则化。
英文摘要
Shock stabilization in compressible Euler flows remains a central challenge for high-order numerical methods. Existing shock-capturing approaches, including limiters, artificial viscosity, and reconstruction-based methods, involve tradeoffs between robustness, accuracy, preservation of fine-scale flow features, and computational complexity. In this work, we develop a discontinuous Galerkin (DG) discretization of the information geometric regularization (IGR) framework introduced by Cao and Schäfer for the compressible Euler equations. The method regularizes shocks at the PDE level by augmenting the Euler equations with the entropic pressure $Σ$, obtained from an auxiliary elliptic equation. Within the DG formulation, the regularization enters only through the augmented pressure $P+Σ$ in the Euler fluxes, preserving the conservative structure of the discretization while using a common approximation space for both the hyperbolic and elliptic equations. Numerical experiments spanning one and two-dimensional benchmark problems show the proposed formulation stabilizes shocks without shock-capturing limiters or artificial viscosity, although positivity-preserving methods may still be required when the density or pressure approaches zero. Compared with a characteristic TVB-limited DG formulation, the IGR-DG method resolves increasingly finer-scale flow features as the polynomial order is increased while maintaining stable shock resolution. The entropic pressure remains localized to regions of strong compression with minimal activation in smooth regions of the flow, providing selective PDE-level regularization while preserving the underlying solution elsewhere.
