Speaker: 夏银华 （中国科学技术大学）
Time: 2025-10-21 15:00-16:00
Location: 1410, SIMIS
Abstract: In this talk, we present a unified and general framework for constructing arbitrary high-order well-balanced discontinuous Galerkin (DG) methods for hyperbolic balance laws, including the shallow water equations (SWEs), the compressible Euler equations with gravity, the Ripa model, and two-layer SWEs. The proposed approach achieves exact preservation of both hydrostatic and moving equilibrium states, encompassing complex equilibria such as the isobaric steady states in the Ripa system. The key idea is to approximate the equilibrium variables—rather than the conservative variables—within the DG polynomial space. This choice is crucial to ensuring the well-balanced property at arbitrary order. To robustly treat nonconservative products arising in systems such as the two-layer shallow water equations, the method incorporates the Dal Maso–LeFloch–Murat (DLM) theory within the DG framework. Our approach offers great flexibility: it is compatible with any consistent numerical flux, and it avoids the need for equilibrium-state reconstruction or special source-term balancing. This allows the development of well-balanced schemes for non-hydrostatic equilibria in Euler-type systems. Extensive numerical examples, such as moving or isobaric equilibria, validate the high-order accuracy and exact equilibrium preservation for various flows given by hyperbolic balance laws. With a relatively coarse mesh, it is also possible to capture small perturbations at or close to steady flow without numerical oscillations.
在本报告中，我们提出了一类高阶平衡间断有限元方法，用于求解双曲型平衡律方程，包括浅水方程、带重力的可压缩欧拉方程以及Ripa模型等。该方法能够精确保持静水平衡态与动水平衡态，并适用于如Ripa系统中的等压稳态等复杂平衡形式。本方法的核心思想是：在间断有限元逼近空间中，对平衡变量而非守恒变量进行多项式近似。该策略对于保证任意高阶的守恒平衡性质至关重要。针对两层浅水方程中出现的非守恒项，本研究在间断有限元框架中引入了Dal Maso–LeFloch–Murat（DLM）理论，以确保数值方法的稳定性与收敛性。该方法具有高度的灵活性：可与任意相容的数值通量结合使用，无需额外的平衡态恢复或特殊的源项处理，从而能够自然地扩展至具有非静水平衡态的更一般双曲系统。通过一系列数值算例（包括动平衡态与等压平衡态等），验证了该方法的高阶精度、严格的平衡保持能力，以及在粗网格条件下对平衡态附近微扰的无振荡捕捉性能，充分展示了其在求解双曲平衡律方程中的高效性与通用性。
About Speaker: 夏银华，中国科学技术大学数学科学学院，教授，博士生导师，安徽省领军人才特聘教授。中国科学技术大学数学系获得博士学位，曾先后到美国布朗大学、香港大学、德国维堡大学等从事博士后研究和访问工作。主要从事高精度数值方法和大规模科学计算的研究，应用于计算流体、天体物理、相场问题、交通流等方面的数值模拟。相关工作发表在包括Math. Comp., J. Comput. Phys., J. Sci. Comput., SIAM J. Num. Anal., SIAM J. Sci. Comput.等杂志。主持国家自然科学基金、教育部、安徽省杰青项目等多项科学基金项目的研究
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