Liu Yang, Zhu Huajun, Yan Zhen-Guo, Jia Feiran, Feng Xinlong
School of Mathematics and Systems Science, Xinjiang University, Urumqi 830017, China.
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang 621000, China.
Entropy (Basel). 2023 Jun 8;25(6):911. doi: 10.3390/e25060911.
The discontinuous Galerkin spectral element method (DGSEM) is a compact and high-order method applicable to complex meshes. However, the aliasing errors in simulating under-resolved vortex flows and non-physical oscillations in simulating shock waves may lead to instability of the DGSEM. In this paper, an entropy-stable DGSEM (ESDGSEM) based on subcell limiting is proposed to improve the non-linear stability of the method. First, we discuss the stability and resolution of the entropy-stable DGSEM based on different solution points. Second, a provably entropy-stable DGSEM based on subcell limiting is established on Legendre-Gauss (LG) solution points. Numerical experiments demonstrate that the ESDGSEM-LG scheme is superior in non-linear stability and resolution, and ESDGSEM-LG with subcell limiting is robust in shock-capturing.
间断伽辽金谱元法（DGSEM）是一种适用于复杂网格的紧凑高阶方法。然而，在模拟欠分辨率涡旋流时的混叠误差以及在模拟激波时的非物理振荡可能会导致DGSEM的不稳定性。本文提出了一种基于子单元限制的熵稳定DGSEM（ESDGSEM），以提高该方法的非线性稳定性。首先，我们讨论了基于不同求解点的熵稳定DGSEM的稳定性和分辨率。其次，在勒让德-高斯（LG）求解点上建立了一种可证明熵稳定的基于子单元限制的DGSEM。数值实验表明，ESDGSEM-LG格式在非线性稳定性和分辨率方面表现优异，并且具有子单元限制的ESDGSEM-LG在激波捕捉方面具有鲁棒性。
