### Abstract

Language | English |
---|---|

Pages | 973-991 |

Number of pages | 19 |

Journal | Journal of Computational Physics |

Volume | 376 |

Early online date | 1 Sep 2018 |

DOIs | |

State | Published - 1 Jan 2019 |

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### Keywords

- hybridizable discontinuous Galerkin
- gas kinetic equation
- synthetic iterative scheme
- fast convergence

### Cite this

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**A high-order hybridizable discontinuous Galerkin method with fast convergence to steady-state solutions of the gas kinetic equation.** / Su, Wei; Wang, Peng; Zhang, Yonghao; Wu, Lei.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A high-order hybridizable discontinuous Galerkin method with fast convergence to steady-state solutions of the gas kinetic equation

AU - Su,Wei

AU - Wang,Peng

AU - Zhang,Yonghao

AU - Wu,Lei

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - hybridizable discontinuous Galerkin

KW - gas kinetic equation

KW - synthetic iterative scheme

KW - fast convergence

UR - https://www.journals.elsevier.com/journal-of-computational-physics

U2 - 10.1016/j.jcp.2018.08.050

DO - 10.1016/j.jcp.2018.08.050

M3 - Article

VL - 376

SP - 973

EP - 991

JO - Journal of Computational Physics

T2 - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -