[论文解读] Performance Enhancement for High-order Gas-kinetic Scheme Based on WENO-adaptive-order Reconstruction

Xing Ji, Kun Xu|arXiv (Cornell University)|May 21, 2019

Computational Fluid Dynamics and Aerodynamics参考文献 46被引用 5

一句话总结

该论文提出了一种基于WENO自适应阶数（WENO-AO）重构的性能增强型高阶气体动理学格式（HGKS），以克服传统HGKS的局限性。通过单次多项式重构同时提供单元界面值和斜率，该方法消除了对单独平衡态重构的需求，将精度提升至理论阶数，减少非物理振荡，从而在1D–3D模拟中实现更高的鲁棒性和效率。

ABSTRACT

High-order gas-kinetic scheme (HGKS) has been well-developed in the past years. Abundant numerical tests including hypersonic flow, turbulence, and aeroacoustic problems, have been used to validate its accuracy, efficiency, and robustness. However, there are still rooms for its further improvement. Firstly, the reconstruction in the previous scheme mainly achieves a third-order accuracy for the initial non-equilibrium states. At the same time, the equilibrium state in space and time in HGKS has to be reconstructed separately. Secondly, it is complicated to get reconstructed data at Gaussian points from the WENO-type method in high dimensions. For HGKS, besides the point-wise values at the Gaussian points it also requires the slopes in both normal and tangential directions of a cell interface. Thirdly, there exists visible spurious overshoot/undershoot at weak discontinuities from the previous HGKS with the standard WENO reconstruction. In order to overcome these difficulties, in this paper we use an improved reconstruction for HGKS. The WENO with adaptive order (WENO-AO) method is implemented for reconstruction.A whole polynomial inside each cell is provided in WENO-AO reconstruction. The HGKS becomes simpler than the previous one with the direct implementation of cell interface values and their slopes from WENO-AO. The additional reconstruction of equilibrium state at the beginning of each time step can be avoided as well by dynamically merging the reconstructed non-equilibrium slopes. The new HGKS essentially releases or totally removes the above existing problems in previous HGKS. The accuracy of the scheme from 1D to 3D from the new HGKS can recover the theoretical order of accuracy of the WENO reconstruction.In the two- and three-dimensional simulations, the new HGKS shows better robustness and efficiency than the previous scheme in all test cases.

研究动机与目标

• 解决以往HGKS因WENO界面值与斜率重构不一致导致的三阶精度限制问题。
• 通过动态合并非平衡态斜率，消除每时间步单独进行平衡态重构的需求。
• 减少标准WENO基HGKS在弱间断附近出现的非物理振荡。
• 通过利用完整的多项式重构，在1D、2D和3D中实现高阶格式的理论阶精度。
• 在复杂流动模拟（如高超声速流和湍流）中提升计算效率和鲁棒性。

提出的方法

• 实施WENO-AO重构，将每个单元内的单元界面值和斜率统一通过单个多项式生成。
• 直接在通量计算中使用重构的斜率和界面值，跳过额外的斜率重构步骤。
• 通过动态合并重构的非平衡态斜率，消除每时间步单独进行平衡态重构的需要。
• 在多阶段多导数（MSMD）框架中使用通量函数的时间导数，实现高阶时间精度。
• 将气体动理学通量函数适配为通过重构多项式同时包含法向和切向导数。
• 在1D、2D和3D中应用GKS通量函数，通过宏观导数的矩方法重构微观分布函数。

实验结果

研究问题

• RQ1 WENO-AO重构是否能在多维情况下恢复高阶HGKS的理论精度？
• RQ2 消除单独的平衡态重构是否能提升计算效率和鲁棒性？
• RQ3 与标准WENO基HGKS相比，新重构策略是否能减少弱间断附近的非物理振荡？
• RQ4 在2D和3D测试案例中，新HGKS在精度、稳定性及计算成本方面与以往方案相比表现如何？
• RQ5 完整多项式重构在多大程度上增强了GKS通量的多维特性？

主要发现

• 新HGKS在1D、2D和3D模拟中均实现了WENO重构的理论精度，且在光滑区域无精度退化。
• 与先前的WENO基HGKS相比，弱间断附近的非物理过冲和欠冲显著减少。
• 由于消除了单独的平衡态重构，计算成本降低，算法简化，效率提升。
• 该格式在高超声速流和湍流等复杂测试案例中表现出卓越的鲁棒性和稳定性。
• 采用WENO-AO重构的两阶段四阶HGKS在所有测试问题中均优于以往版本，在精度、效率和鲁棒性方面表现更优。
• 该方法在复杂多尺度流动问题（如非平衡多温度流）中仍能保持高阶精度和稳定性。