AI 中文总结
本文针对双曲守恒律，在一般非结构网格上建立半离散熵稳定间断Galerkin方法的严格先验误差估计，证明其收敛阶，还扩展至熵稳定无振荡DG方法，数值实验显示收敛率或超理论预测。
AI 中文摘要
熵不等式是双曲守恒律方程适定性的基础，为从无穷多弱解中选取物理解提供了必要准则。Chen和Shu[J. Comput. Phys. 345 (2017)]提出了一种统一框架，通过特定数值求积构造满足任意给定熵的高阶间断Galerkin（DG）方法；然而，其伴随的误差分析仅局限于截断误差水平，在这类熵稳定格式的严格收敛理论中留下了关键缺口。本文通过对一般非结构网格上半离散熵稳定DG（ESDG）方法建立严格先验误差估计，填补了该缺口，该分析适用于标量方程和方程组，且基于直接在节点层面开展的有限差分型相容-稳定性论证。在多项式重构假设和L∞先验界下，我们证明了基于求积的范数（该范数在有限维重构空间上等价于分片L²范数）下的O(hᵏ)误差估计。我们进一步将该框架扩展至Liu、Lu和Shu[SIAM J. Sci. Comput. 46 (2024)]提出的熵稳定无振荡DG（ESOFDG）方法，证明额外阻尼项不会降低收敛阶。数值实验表明，观测到的收敛率可能超出理论预测最多半个阶。
英文摘要
Entropy inequalities are fundamental to the well-posedness of hyperbolic conservation laws, providing the essential criterion for selecting the physically admissible solution among infinitely many weak solutions. Chen and Shu [J. Comput. Phys. 345 (2017)] proposed a unified framework for constructing high-order discontinuous Galerkin (DG) methods that satisfy entropy inequalities for any given entropy via specific numerical quadrature; however, their accompanying error analysis was limited to the truncation error level, leaving a critical gap in the rigorous convergence theory for these entropy-stable schemes. This paper closes that gap by establishing rigorous a priori error estimates for semi-discrete entropy-stable DG (ESDG) methods on general unstructured meshes for hyperbolic conservation laws. The analysis applies to both scalar equations and systems, and is built upon a finite-difference-type consistency-stability argument carried out directly at the nodal level. Under a polynomial-reconstruction hypothesis and an $L^\infty$ a priori bound, we prove an $O(h^k)$ error estimate in a quadrature-based norm, which is equivalent to the broken $L^2$ norm on the finite-dimensional reconstruction space. We further extend this framework to the entropy-stable oscillation-free DG (ESOFDG) method introduced by Liu, Lu, and Shu [SIAM J. Sci. Comput. 46 (2024)], demonstrating that the additional damping terms do not degrade the convergence order. Numerical experiments suggest that the observed convergence rates may exceed the theoretical prediction by up to half an order.
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