双曲守恒律的主动通量方法——通量向量分裂与保界：二维情形
Active flux methods for hyperbolic conservation laws -- flux vector splitting and bound-preservation: Two-dimensional case
AI总结：
针对二维双曲守恒律，本文拓展主动通量方法，提出基于通量向量分裂的点值更新与激波传感器限制器，验证了其精度、保界性与激波捕捉能力，在高马赫数流动中兼具效率与潜力。
摘要：
本文研究二维双曲守恒律的主动通量（active flux, AF）方法，重点关注用于点值更新的通量向量分裂（flux vector splitting, FVS）与保界（bound-preserving, BP）限制器，是对我们前期工作[J.M. Duan, W. Barsukow, C. Klingenberg, arXiv:2405.02447]的拓展。研究表明，基于FVS的点值更新可解决笛卡尔网格上沿单轴方向的准二维黎曼问题中出现的网格对齐问题。因此，基于FVS的AF方法优于采用雅可比分裂的AF方法，后者易出现跨声速与网格对齐问题。本文提出一种基于激波传感器的限制器来改进单元平均值的凸限制，可有效降低振荡。通过若干基准问题测试验证了所提BP AF方法的精度、保界性质与激波捕捉能力。此外，针对双马赫反射与前向台阶问题，在相同网格分辨率下，本文AF方法相比带TVB限制器的三阶间断Galerkin方法，能捕捉到相当或更优的小尺度特征，且使用的自由度更少，证明了BP AF方法用于高马赫数流动的效率与潜力。
英文摘要：
This paper studies the active flux (AF) methods for two-dimensional hyperbolic conservation laws, focusing on the flux vector splitting (FVS) for the point value update and bound-preserving (BP) limitings, which is an extension of our previous work [J.M. Duan, W. Barsukow, C. Klingenberg, arXiv:2405.02447]. The FVS-based point value update is shown to address the mesh alignment issue that appeared in a quasi-2D Riemann problem along one axis direction on Cartesian meshes. Consequently, the AF methods based on the FVS outperform those using Jacobian splitting, which are prone to transonic and mesh alignment issues. A shock sensor-based limiting is proposed to enhance the convex limiting for the cell average, which can reduce oscillations well. Some benchmark problems are tested to verify the accuracy, BP property, and shock-capturing ability of our BP AF method. Moreover, for the double Mach reflection and forward-facing step problems, the present AF method can capture comparable or better small-scale features compared to the third-order discontinuous Galerkin method with the TVB limiter on the same mesh resolution, while using fewer degrees of freedom, demonstrating the efficiency and potential of our BP AF method for high Mach number flows.
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