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1. 三峡大学 水利与环境学院, 宜昌 443002;
2. 北京航空航天大学 航空科学与工程学院, 北京 100083

Adaptive simple WENO limiter-discontinuous Galerkin method for Euler equations
WU Zeyan1, WANG Lifeng2 , WU Zhe2
1. College of Hydraulic and Environmental Engineering, China Three Gorges University, Yichang 44300;
2. School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2015-04-18; Accepted: 2015-07-17; Published online: 2015-10-14 15:18
Foundation item: Talents Scientific Research Starting Foundation of China Three Gorges University (KJ2014B031)
Corresponding author. Tel.: 010-82339560 E-mail: wanglifeng@ase.buaa.edu.cn
Abstract: To achieve high precision and high resolution numerical result of Euler equations, the basic principle of discontinuous Galerkin method, the simple WENO limiter on triangular meshes and shock capturing method based on adaptive mesh refinement were introduced. The simple WENO limiter-discontinuous Galerkin method was applied to the curved quadrilateral element, and the adjacent elements of every element with the same coordinates of the Gauss integral points on the boundaries were found. The adaptive computation based on “trouble element” refinement was accomplished. Several benchmark test cases were computed. The numerical results show that the simple WENO limiter is appropriate for the curvilinear boundary quadrilateral element and for the shock capturing based on unstructured grids with hanging nodes.
Key words: discontinuous Galerkin method     simple WENO limiter     Euler equations     adaptive computation     curvilinear boundary quadrilateral element

DG方法是有限元方法与有限体积方法的结合,在基函数和检验函数的运用方面与有限元方法相似,而在对方程性质的近似方面与有限体积方法更接近,因此有若干优点。首先,与有限元方法一样,间断有限元方法可以方便地提高空间和时间方向的计算精度。在空间上,DG方法可以方便地实施自适应计算,其有2个方面的含义:①对网格单元的自适应加密。DG方法允许单元间出现“悬挂节点”,因此网格加密非常简单。②DG方法允许不同的单元采用不同阶的多项式。这2个方面分别称作h自适应和p自适应。其次,由DG方法形成的常微分方程组,其质量矩阵是分块对角的,甚至可以是单位矩阵,只要在各个单元内选择合适的正交多项式基函数。因为分块对角矩阵求逆相当于各个块矩阵的求逆,所以质量矩阵的求逆计算量极小。分块对角矩阵的逆仍然是分块对角矩阵,所以存储量也极小。最后,间断有限元方法特别适于并行计算。从已有文献可知,间断有限元方法的并行计算效率在80%以上[6]

1 数学模型

2 空间及时间离散 2.1 DG空间离散的一般理论

2.2 时间离散

3 简单WENO限制器

1) 修改每一个pl,使得 在“问题单元”Δ0上具有与p0(x,y)相同的单元平均。

2) 记线性加权系数为ξ0ξ1ξ2ξ3,因为在光滑区域,4个多项式 l(x,y)(l=0,1,2,3)都具有k+1阶精度,所以对于加权系数,首先要求满足ξ0+ξ1+ξ2+ξ3=1。另外,通过调节加权系数ξ0ξ1ξ2ξ3来满足解的精度与激波处的本质无振荡的平衡。一般地,取ξ0=0.997,ξ1=ξ2=ξ3=0.001。

3) 计算光滑指示子βl,l=0,1,2,3。光滑指示子显式了函数 l(x,y)在“问题单元”Δ0上的光滑度。取光滑指示子为

4) 计算基于光滑指示子的非线性加权系数ωl,l=0,1,2,3。

5) 非线性WENO限制器重构多项式( )inew(x,y)定义为 i(x,y)(i=0,1,2,3)的凸组合。

6) 将( )inew(x,y)投影回物理空间得到

4 基于自适应网格加密的激波捕捉

Euler方程是强非线性方程,即使初值连续,其解也可能出现间断。为了捕捉间断,一般会在计算中引入人工黏性,或者用自带耗散效应的格式(如迎风格式、WENO重构等)来抑制振荡。这时,一般只能在1～5个网格内捕捉到激波,计算的分辨率比较低。所以,要提高间断的分辨率,就需要非常细的网格。如果在全流场采用一致的细网格,无疑会增加内存消耗和计算量,也意味着增加了计算时间,降低了计算效率。为了有效解决这个问题,依据解的变化来动态调整网格疏密程度是必要的,这就是通常的h自适应方法。这里简单介绍本文采用的h自适应方法。

4.1 单元间的数据交换

4.2 加密/粗化单元的标识

4.3 加密/粗化过程

5 数值算例 5.1 双马赫反射

 图 1 不同时刻密度等值线 Fig. 1 Density contours at different monents

5.2 NACA0012翼型跨声速绕流

 图 2 NACA0012翼型跨声速绕流原始网格(局部) Fig. 2 Local view of original mesh for transonic NACA0012 flow
 图 3 NACA0012翼型跨声速绕流局部加密网格 Fig. 3 Local view of adaptive mesh for transonic NACA0012 flow
 图 4 NACA0012翼型跨声速绕流原始网格下的 马赫数等值线 Fig. 4 Mach number contours on original mesh for transonic NACA0012 flow
 图 5 NACA0012翼型跨声速绕流网格局部加密后的马赫数等值线 Fig. 5 Mach number contours on adaptive mesh for transonic NACA0012 flow
 图 6 NACA0012翼型跨声速绕流原始网格下的表面压力系数分布 Fig. 6 Surface pressure coefficient distribution on original mesh for transonic NACA0012 flow
 图 7 NACA0012翼型跨声速绕流网格局部加密后的表面压力系数分布 Fig. 7 Surface pressure coefficient distribution on adaptive mesh for transonic NACA0012 flow
5.3 4%凸起

 图 8 4%凸起原始网格 Fig. 8 Full view of original mesh for subsonic flow past 4% bump configuration
 图 9 4%凸起局部加密网格 Fig. 9 Full view of adaptive mesh for subsonic flow past 4% bump configuration
 图 10 4%凸起原始网格下的密度等值线 Fig. 10 Density contours on original mesh for subsonic flow past 4% bump configuration
 图 11 4%凸起局部网格加密后的密度等值线 Fig. 11 Density contours on adaptive mesh for subsonic flow past 4% bump configuration
5.4 2段NACA0012翼型超声速绕流

 图 12 2段NACA0012翼型原始网格(局部) Fig. 12 Local view of original mesh for supersonic flow through two staggered NACA0012 airfoils
 图 13 2段NACA0012翼型局部加密网格 Fig. 13 Local view of adaptive mesh for supersonic flow through two staggered NACA0012 airfoils
 图 14 2段NACA0012翼型原始网格下的 马赫数等值线 Fig. 14 Mach number contours on original mesh for supersonic flow through two staggered NACA0012 airfoils
 图 15 2段NACA0012翼型局部网格加密后的马赫数等值线 Fig. 15 Mach number contours on adaptive mesh for supersonic flow through two staggered NACA0012 airfoils
6 结 论

1) 简单WENO限制器可以推广应用到曲边四边形单元上。

2) 简单WENO限制器可适用于局部网格加密时具有“悬挂节点”的非结构网格上的激波捕捉。

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#### 文章信息

WU Zeyan, WANG Lifeng, WU Zhe

Adaptive simple WENO limiter-discontinuous Galerkin method for Euler equations

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(4): 806-814.
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0237