Fast numerical method for the Boltzmann equation on non-uniform grids

Heintz, Alexei; Kowalczyk, Piotr; Grzhibovskis, Richards

doi:10.1016/j.jcp.2008.03.028

Cited by 10 publications

(3 citation statements)

References 24 publications

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“…The dealiasing condition requires the velocity domain to be about two times larger than the support of the distribution function, which wastes more than half of the compute memory and time in three-dimensional velocity space. A better way to do this is to introduce the nonuniform FFT [53], where the velocity space is non-uniformly discretized but the frequency space is equally divided. The only change we need is to get the spectrumf from f and the collision operator Q fromQ by non-uniform FFTs, while the FFT-based convolution remains unchanged.…”

Section: Discussionmentioning

confidence: 99%

Deterministic numerical solutions of the Boltzmann equation using the fast spectral method

White

Scanlon

et al. 2013

Journal of Computational Physics

148

165

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The Boltzmann equation describes the dynamics of rarefied gas flows, but the multidimensional nature of its collision operator poses a real challenge for its numerical solution. In this paper, the fast spectral method [36], originally developed by Mouhot and Pareschi for the numerical approximation of the collision operator, is extended to deal with other collision kernels, such as those corresponding to the soft, Lennard–Jones, and rigid attracting potentials. The accuracy of the fast spectral method is checked by comparing our numerical solutions of the space-homogeneous Boltzmann equation with the exact Bobylev–Krook–Wu solutions for a gas of Maxwell molecules. It is found that the accuracy is improved by replacing the trapezoidal rule with Gauss–Legendre quadrature in the calculation of the kernel mode, and the conservation of momentum and energy are ensured by the Lagrangian multiplier method without loss of spectral accuracy. The relax-to-equilibrium processes of different collision kernels with the same value of shear viscosity are then compared; the numerical results indicate that different forms of the collision kernels can be used as long as the shear viscosity (not only the value, but also its temperature dependence) is recovered. An iteration scheme is employed to obtain stationary solutions of the space-inhomogeneous Boltzmann equation, where the numerical errors decay exponentially. Four classical benchmarking problems are investigated: the normal shock wave, and the planar Fourier/Couette/force-driven Poiseuille flows. For normal shock waves, our numerical results are compared with a finite difference solution of the Boltzmann equation for hard sphere molecules, experimental data, and molecular dynamics simulation of argon using the realistic Lennard–Jones potential. For planar Fourier/Couette/force-driven Poiseuille flows, our results are compared with the direct simulation Monte Carlo method. Excellent agreements are observed in all test cases, demonstrating the merit of the fast spectral method as a computationally efficient method for rarefied gas dynamics

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Section: Discussionmentioning

confidence: 99%

Deterministic numerical solutions of the Boltzmann equation using the fast spectral method

White

Scanlon

et al. 2013

Journal of Computational Physics

148

165

View full text Add to dashboard Cite

show abstract

“…Simulation of gas mixtures and gases with internal energies, as well as multidimensional models can be found in [14][15][16][17][18][19][20], and references therein. Other fast methods include representing the solution as a sum of homogeneous Gaussians [21,22], polynomial spectral discretization [23], utilizing non-uniform meshes [24], and a hyperbolic cross approximation [25]. Additional review of recent results can be found in [26,27].…”

Section: Introductionmentioning

confidence: 99%

Acceleration of Boltzmann Collision Integral Calculation Using Machine Learning

2021

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The Boltzmann equation is essential to the accurate modeling of rarefied gases. Unfortunately, traditional numerical solvers for this equation are too computationally expensive for many practical applications. With modern interest in hypersonic flight and plasma flows, to which the Boltzmann equation is relevant, there would be immediate value in an efficient simulation method. The collision integral component of the equation is the main contributor of the large complexity. A plethora of new mathematical and numerical approaches have been proposed in an effort to reduce the computational cost of solving the Boltzmann collision integral, yet it still remains prohibitively expensive for large problems. This paper aims to accelerate the computation of this integral via machine learning methods. In particular, we build a deep convolutional neural network to encode/decode the solution vector, and enforce conservation laws during post-processing of the collision integral before each time-step. Our preliminary results for the spatially homogeneous Boltzmann equation show a drastic reduction of computational cost. Specifically, our algorithm requires O(n3) operations, while asymptotically converging direct discretization algorithms require O(n6), where n is the number of discrete velocity points in one velocity dimension. Our method demonstrated a speed up of 270 times compared to these methods while still maintaining reasonable accuracy.

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“…A survey of this subject can be found in [19]. While the majority of authors use an uniform grid in the velocity space, in [25] A. Heintz, P, Kowalczyk and R. Grzhibovskis have used a non-uniform grid.…”

Section: Introductionmentioning

confidence: 99%

Galerkin–Petrov approach for the Boltzmann equation

Gamba¹,

Rjasanow²

2018

Journal of Computational Physics

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In this work, we propose a new Galerkin-Petrov method for the numerical solution of the classical spatially homogeneous Boltzmann equation. This method is based on an approximation of the distribution function by associated Laguerre polynomials and spherical harmonics and test an a variational manner with globally defined threedimensional polynomials. A numerical realization of the algorithm is presented. The algorithmic developments are illustrated with the help of several numerical tests.

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Fast numerical method for the Boltzmann equation on non-uniform grids

Cited by 10 publications

References 24 publications

Deterministic numerical solutions of the Boltzmann equation using the fast spectral method

Deterministic numerical solutions of the Boltzmann equation using the fast spectral method

Acceleration of Boltzmann Collision Integral Calculation Using Machine Learning

Galerkin–Petrov approach for the Boltzmann equation

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