A high-order hybridizable discontinuous Galerkin method with fast convergence to steady-state solutions of the gas kinetic equation
A high-order hybridizable discontinuous Galerkin method with fast convergence to steady-state solutions of the gas kinetic equation
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DOI：
10.1016/j.jcp.2018.08.050
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发表时间：
2018-03
期刊：
影响因子：
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通讯作者：
Wei-Jen Su;Peng Wang;Yonghao Zhang;Lei Wu
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文献类型：
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作者：
Wei-Jen Su;Peng Wang;Yonghao Zhang;Lei Wu
在多孔介质中气体运移的孔网模拟中，稀薄气体通过任意形状的二维长管的Poiseuille流动的质量流量是至关重要的。本文首次采用高阶可杂交间断Galerkin(HDG)方法求解二维三角网格上线性化的Bhatnagar-Gross-Krook方程的稳态解。在三角形网格和网格骨架上，速度分布函数及其迹线分别在分段多项式空间(最高4次)中逼近。通过采用一阶迎风格式得到的数值通量，并将其连续性弱地施加到网格骨架上，得到了与原始不连续Galerkin格式相比具有更少耦合自由度的未知迹线的整体系统。为了达到快速收敛到定态解的目的，在相同的网格上，用HDG同时求解了一个渐近保持到流体动力极限的类扩散速度方程。实验证明，本文提出的HDG合成迭代格式具有较高的精度和效率。具体地说，对于近连续区域的流动，数值模拟表明，在达到相同精度的情况下，我们的格式可以比传统的迭代格式快两个数量级，并且比基于空间有限差分离散的合成迭代格式快一个数量级。此外，隐式HDG方法比显式间断Galerkin气体动力学求解器以及逼近次数大于2时的隐式间断Galerkin格式更有效。HDG合成迭代格式已准备好扩展到模拟稀薄混合气体和Boltzmann碰撞算符。
The mass flow rate of Poiseuille flow of rarefied gas through long ducts of two-dimensional cross-sections with arbitrary shape is critical in the pore-network modeling of gas transport in porous media. Here, for the first time, the high-order hybridizable discontinuous Galerkin (HDG) method is used to find the steady-state solution of the linearized Bhatnagar–Gross–Krook equation on two-dimensional triangular meshes. The velocity distribution function and its traces are approximated in piecewise polynomial spaces (of degree up to 4) on the triangular meshes and mesh skeletons, respectively. By employing a numerical flux that is derived from the first-order upwind scheme and imposing its continuity weakly on the mesh skeletons, global systems for unknown traces are obtained with fewer coupled degrees of freedom when compared to the original discontinuous Galerkin formulation. To achieve fast convergence to the steady-state solution, a diffusion-like equation for flow velocity, which is asymptotic-preserving into the fluid dynamic limit, is solved by the HDG simultaneously on the same meshes. The proposed HDG-synthetic iterative scheme is proved to be accurate and efficient. Specifically, for flows in the near-continuum regime, numerical simulations have shown that, to achieve the same level of accuracy, our scheme could be faster than the conventional iterative scheme by two orders of magnitude, also it is faster than the synthetic iterative scheme based on the finite difference discretization in the spatial space by one order of magnitude. In addition, the implicit HDG method is more efficient than an explicit discontinuous Galerkin gas kinetic solver, as well as the implicit discontinuous Galerkin scheme when the degree of approximating polynomial is larger than 2. The HDG-synthetic iterative scheme is ready to be extended to simulate rarefied gas mixtures and the Boltzmann collision operator.