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

Abstract

The mass flow rate of Poiseuille flow of rarefied gas through long ducts of two-dimensional cross-sections with arbitrary shape are critical in the pore-network modeling of gas transport in porous media. In this paper, 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 the piecewise polynomial space (of degree up to 4) on the triangular meshes and the mesh skeletons, respectively. By employing a numerical flux that is derived from the first-order upwind scheme and imposing its continuity on the mesh skeletons, global systems for unknown traces are obtained with a few coupled degrees of freedom. To achieve fast convergence to the steady-state solution, a diffusion-type equation for flow velocity that 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, while it is faster than the synthetic iterative scheme based on the finite difference discretization in the spatial space by one order of magnitude. The HDG-synthetic iterative scheme is ready to be extended to simulate rarefied gas mixtures and the Boltzmann collision operator.

Rarefied flow separation in microchannel with bends

“…For β = 0.99999, the DVM simulation converges very slowly and requires a very fine spatial grid at such low Kn (Valougeorgis & Naris 2003;Wang et al 2018). Therefore, the discontinuous Galerkin method (DGM) and general synthetic iterative scheme, which enables the fast convergence to the steady-state solution and retains asymptotic preserving natures of the Navier-Stokes equations, is employed to solve the linearised Shakhov equation for the case of pressure ratio β = 0.99999 (Su et al 2019b(Su et al , 2020b. The DVM and DGM solutions are compared with each other and with the other available data, see appendix A.…”

Can we find steady-state solutions to multiscale rarefied gas flows within dozens of iterations?

“…where the derivative with respect to x can be approximated by any conventional CFD schemes such as the finite difference, finite volume, or Discontinuous Galerkin (DG) methods [41,42], and the collision operator in Eq. (4) can be calculated by the fast spectral method [37,39] based on the velocity distribution function at the k-th iteration step.…”

“…by a second-order upwind finite difference in the bulk and a first-order upwind scheme at the solid surface [40] or the DG method [41,42].…”

“…Since the gas kinetic equation is solved together with the macroscopic equations (13), (17) and (20) for general rarefied gas flows, the above scheme is called the GSIS. Note that although the SIS has been widely applied to the radiation transport processes [23] and rarefied gas flows driven by local pressure, temperature, and concentration gradients [43,29,50,51,41] to overcome the slow convergence in the near-continuum flow regime, it is the first time that the GSIS is developed for general rarefied gas flows described by the LBE. Also, it is with no doubt that such a methodology can be directly applied to construct the GSIS for the nonlinear Boltzmann equation.…”

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

“…More recently, special attention was dedicated to the development of positivity-preserving Riemann solvers in the context of hybridised DG methods [263]. For a complete review on HDG methods for compressible flows, interested readers are referred to [263], whereas the application to gas kinetics modelled by means of the linearised Bhatnagar-Gross-Krook equation is discussed in [254].…”