Deterministic solution of the spatially homogeneous Boltzmann equation using discontinuous Galerkin discretizations in the velocity space

Alekseenko, Alexander; Josyula, Eswar

doi:10.1016/j.jcp.2014.03.031

Cited by 35 publications

(63 citation statements)

References 40 publications

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“…These methods has been applied to spatially inhomogeneous problems in [29,28,10]. Other methods include the fast discrete velocity method [21] and the discontinuous Galerkin method [1]. Another type of spectral method based on global orthogonal polynomials is also being studied recently [8,13,25].…”

Section: Introductionmentioning

confidence: 99%

Burnett spectral method for the spatially homogeneous Boltzmann equation

2020

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We develop a spectral method for the spatially homogeneous Boltzmann equation using Burnett polynomials in the basis functions. Using the sparsity of the coefficients in the expansion of the collision term, the computational cost is reduced by one order of magnitude for general collision kernels and by two orders of magnitude for Maxwell molecules. The proposed method can couple seamlessly with the BGK-type modelling techniques to make future applications affordable. The implementation of the algorithm is discussed in detail, including a numerical scheme to compute all the coefficients accurately, and the design of the data structure to achieve high cache hit ratio. Numerical examples are provided to demonstrate the accuracy and efficiency of our method.

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Section: Introductionmentioning

confidence: 99%

Burnett spectral method for the spatially homogeneous Boltzmann equation

2020

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“…Great efforts have been devoted to overcoming the above limitations in various aspects. In addition to the commonly used techniques such as high-order discretization scheme or automatically adaptive refinement in the spatial and velocity spaces [12,13,14], two alternative approaches are worth mentioning here. One is proposed to handle the streaming and collision simultaneously so that the restriction on cell size and time step could be significantly relaxed.…”

Section: Introductionmentioning

confidence: 99%

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

Wu¹,

Wang²,

Zhang³

et al. 2019

Journal of Computational Physics

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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.

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“…We argue in this paper that this convolution form leads to development of efficient discretizations of the collision operator using structured locally supported bases. We present a numerical approach that is based on high order nodal discontinuous Galerkin (DG) discretizations of the Boltzmann equation in the velocity variable [2] and that requires O(N 2 ) operations to evaluate the collision operator.…”

Section: Introductionmentioning

confidence: 99%

“…Approaches based on DG discretizations of the Boltzmann equation in the velocity variable were proposed in [26,1,2,17]. High order DG bases are well suited for approximating discontinuous and high gradient solutions.…”

Section: Introductionmentioning

confidence: 99%

Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{O}(N^2) $\end{document}</tex-math></inline-formula> operations using the discrete fourier transform

Alekseenko¹,

Limbacher²

2019

Kinetic &Amp; Related Models

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We present a numerical algorithm for evaluating the Boltzmann collision operator with O(N 2 ) operations based on high order discontinuous Galerkin discretizations in the velocity variable. To formulate the approach, Galerkin projection of the collision operator is written in the form of a bilinear circular convolution. An application of the discrete Fourier transform allows to rewrite the six fold convolution sum as a three fold weighted convolution sum in the frequency space. The new algorithm is implemented and tested in the spatially homogeneous case, and results in a considerable improvement in speed as compared to the direct evaluation. Simultaneous and separate evaluations of the gain and loss terms of the collision operator were considered. Less numerical error was observed in the conserved quantities with simultaneous evaluation.2000 Mathematics Subject Classification. 76P05,76M10,65M60.

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Deterministic solution of the spatially homogeneous Boltzmann equation using discontinuous Galerkin discretizations in the velocity space

Cited by 35 publications

References 40 publications

Burnett spectral method for the spatially homogeneous Boltzmann equation

Burnett spectral method for the spatially homogeneous Boltzmann equation

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

Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{O}(N^2) $\end{document}</tex-math></inline-formula> operations using the discrete fourier transform

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