标题： 统一界面通量评估在广义间断伽辽金谱单元框架中 显示英文标题

标题： Unified interface flux evaluation in a general discontinuous Galerkin spectral element framework

摘要： 高阶间断伽辽金谱元方法（DGSEM）近年来受到了越来越多的关注和发展，尤其是在计算流体力学领域。 间断伽辽金方法在处理非共形界面（如移动几何或hp-细化中遇到的界面）时固有的灵活性，为实际模拟提供了显著优势。 尽管DG方法有着成熟的数学框架，但为了提高性能和能力，仍然存在实际实现上的挑战。 大多数先前的研究仅关注于结构网格中元素形状或基函数类型的选择，尽管这些研究展示了DGSEM在复杂流动模拟中的能力。 本文讨论了适用于无结构网格中一般谱元的低成本且统一的界面通量评估方法及其在开源谱元框架Nektar++中的实现。 最初的动机源于使用对称内罚法离散Helmholtz方程，其中如果通量在非共形界面上没有正确评估，系统矩阵很容易变得非对称。 本文专注于多项式非共形情况，但理论上可以扩展到几何非共形情况。 还包括不同方法、权衡以及我们初始矩阵自由实现的基准性能比较，这为高性能谱元方法的实现提供了更广泛的讨论内容。 显示英文摘要

摘要： High-order discontinuous Galerkin spectral element methods (DGSEM) have received growing attention and development, especially in the regime of computational fluid dynamics in recent years. The inherent flexibility of the discontinuous Galerkin approach in handling non-conforming interfaces, such as those encountered in moving geometries or hp-refinement, presents a significant advantage for real-world simulations. Despite the well-established mathematical framework of DG methods, practical implementation challenges persist to boost performance and capability. Most previous studies only focus on certain choices of element shape or basis type in a structured mesh, although they have demonstrated the capability of DGSEM in complex flow simulations. This work discusses the low-cost and unified interface flux evaluation approaches for general spectral elements in unstructured meshes, alongside their implementations in the open-source spectral element framework, Nektar++. The initial motivation arises from the discretization of Helmholtz equations by the symmetric interior penalty method, in which the system matrix can easily become non-symmetric if the flux is not properly evaluated on non-conforming interfaces. We focus on the polynomial non-conforming case in this work but extending to the geometric non-conforming case is theoretically possible. Comparisons of different approaches, trade-offs, and performance of benchmark of our initial matrix-free implementation are also included, contributing to the broader discourse on high-performance spectral element method implementations.