Stabilization of the Lattice Boltzmann Method Using Information Theory

Wilson, Tyler L; Pugh, Mary; Dawson, Francis

doi:10.48550/arxiv.1801.03863

Cited by 2 publications

(6 citation statements)

References 61 publications

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“…Consistent with the above analysis, all moments except for J are globally well conserved. This figure also demonstrates that the temporal variation of J per timestep is at most of order 7 .…”

Section: Burgers'mentioning

confidence: 61%

“…This is known to play an essential role for stability in numerical simulations. Indeed, in many previous studies [5][6][7], numerical schemes equipped with the H-theorem exhibited improved robustness. In this study, the main dynamics for ρ is at order t 4 as shown in eq.…”

Section: Burgers'mentioning

confidence: 98%

“…Therefore, it is appropriate to add a condition f v 4 dv = M 0 . Following previous studies [5][6][7], we define H as…”

Section: Burgers'mentioning

confidence: 99%

“…A second key advantage of the kinetic-theoretical approach is that kinetic equations can often be equipped with an H-theorem, or Lyapunov functional. In the lattice Boltzmann approach to fluid dynamics, for example, this feature has been used to develop numerically stable algorithms for the simulation of hydrodynamics [5][6][7]. More generally, even for the study of continuum field theories, it is often easier to find an H-theorem for its microscopic kinetic realization than for its macroscopic hydrodynamic realization.…”

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confidence: 99%

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A kinetic generator for classical field theories with conservation laws

Otomo

Boghosian

Succi

2020

EPL

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In this study, we demonstrate that, by suitably modifying the equilibrium distribution, the Bhatnagar-Gross-Krook form of the Boltzmann equation can serve as a generator of a broad class of nonlinear partial differential equations via a suitable generalization of the Chapman-Enskog method. We achieve this by properly choosing the relaxation time and moments of the equilibrium state in such a way as to suppress the dynamics of the zeroth moment, density, at increasingly faster time scales. As concrete examples, explicit analytical formulations of the Burgers', Korteweg-de Vries and Kuramoto-Sivashinsky equations are presented, along with a discussion of the associated conservation laws of the corresponding moments. The concept is numerically demonstrated by solving the BGK Boltzmann equation with the finite-element method. Because the Boltzmann-BGK equation can be equipped with a H-theorem, the framework developed in this work indicates a path towards the development of new algorithms for the numerical solution of the aforementioned equations with enhanced stability.

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Section: Burgers'mentioning

confidence: 61%

Section: Burgers'mentioning

confidence: 98%

“…Therefore, it is appropriate to add a condition f v 4 dv = M 0 . Following previous studies [5][6][7], we define H as…”

Section: Burgers'mentioning

confidence: 99%

mentioning

confidence: 99%

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A kinetic generator for classical field theories with conservation laws

Otomo

Boghosian

Succi

2020

EPL

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“…On the other hand, the EMRT model is like a "best-effort method": it fixes relaxation parameters up to the second-order moments and then does its best to make the post-collision state's entropy as large as possible. Wilson et al [13] also provided another interpretation of the entropy function based on the "Principle of Minimum Discrimination Information". They derived a similarly constrained minimization of the Kullback-Leibler Divergence problem.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic method for entropic multiple relaxation time model in lattice Boltzmann method

Tang

Öztekin

2022

Phys. Rev. E

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To improve the numerical stability of the lattice Boltzmann method, Karlin et al. [I. V. Karlin, F. Bösch, and S. S. Chikatamarla, Phys. Rev. E 90, 031302(R) (2014)] proposed the entropic multiple relaxation time (EMRT) collision model. The idea behind EMRT is to construct an optimal postcollision state by maximizing its local entropy value. The critical step of the EMRT model is to solve the entropy maximization problem under certain constraints, which is often computationally expensive and even not feasible. In this paper, we propose to employ the perturbation theory and obtain an asymptotic solution to the maximum entropy state. With mathematical analysis of particular cases under relaxed constraints, we obtain the unperturbed form of the original problem and derive the asymptotic solution. We show that the asymptotic solution well approximates the optimal states; thus, our approach provides a novel and efficient way to solve the constraint maximum entropy problem in the EMRT model. Also, we use the same idea of the EMRT model for the initial condition of the distribution function and propose to leave the entropy function to determine the missing information at the initial nodes. Finally, we numerically verify that the simulation results of the EMRT model obtained via the perturbation theory agree well with the exact solution to the Taylor-Green vortex problem. Furthermore, we also demonstrate that the EMRT model exhibits excellent stability performance for under-resolved simulations in the doubly periodic shear layer flow problem.

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Stabilization of the Lattice Boltzmann Method Using Information Theory

Cited by 2 publications

References 61 publications

A kinetic generator for classical field theories with conservation laws

A kinetic generator for classical field theories with conservation laws

Asymptotic method for entropic multiple relaxation time model in lattice Boltzmann method

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