Hyperbolic moment equations using quadrature-based projection methods

Koellermeier, Julian; Torrilhon, Manuel

doi:10.1063/1.4902651

Cited by 16 publications

(14 citation statements)

References 7 publications

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“…However, similar to the moment method in kinetic theory [2,24,39], hyperbolicity of the SWME is limited to a bounded domain in phase space. This lack of hyperbolicity was not discussed in detail in the shallow water moment model [27].…”

Section: Introductionmentioning

confidence: 99%

Analysis and Numerical Simulation of Hyperbolic Shallow Water Moment Equations

Rominger¹,

Marvin²

2020

CiCP

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Shallow Water Moment Equations allow for vertical changes in the horizontal velocity, so that complex shallow flows can be described accurately. However, we show that these models lack global hyperbolicity and the loss of hyperbolicity already occurs for small deviations from equilibrium. This leads to instabilities in a numerical test case. We then derive new Hyperbolic Shallow Water Moment Equations based on a modification of the system matrix. The model can be written in analytical form and hyperbolicity can be proven for a large number of equations. A second variant of this model is obtained by generalizing the modification with the help of additional parameters. Numerical tests of a smooth periodic problem and a dam break problem using the new models yield accurate and fast solutions while guaranteeing hyperbolicity.

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Section: Introductionmentioning

confidence: 99%

Analysis and Numerical Simulation of Hyperbolic Shallow Water Moment Equations

Rominger¹,

Marvin²

2020

CiCP

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“…Lastly, investigation of other collision terms or types of equations might lead to faster simulations for other application cases, too. derived in [39] and written in explicit form first in [41] forp = p 1 + p 2 ,q 1 = 3f 3,0 + f 1,2 andq 2 = 3f 0,3 + f 2,1 .…”

Section: Discussionmentioning

confidence: 99%

“…These conditions ensure that the distribution function has the correct density, momentum and energy. With the chosen basis functions (17), the compatibility conditions can be simplified according to [39] and read…”

Section: Bgk Collision Operatormentioning

confidence: 99%

Projective integration schemes for hyperbolic moment equations

Koellermeier¹,

Samaey²

2021

KRM

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In this paper, we apply projective integration methods to hyperbolic moment models of the Boltzmann equation and the BGK equation, and investigate the numerical properties of the resulting scheme. Projective integration is an explicit scheme that is tailored to problems with large spectral gaps between slow and (one or many) fast eigenvalue clusters of the model. The spectral analysis of a linearized moment model clearly shows spectral gaps and reveals the multi-scale nature of the model for which projective integration is a matching choice. The combination of the non-intrusive projective integration method with moment models allows for accurate, but efficient simulations with significant speedup, as demonstrated using several 1D and 2D test cases with different collision terms, collision frequencies and relaxation times.

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“…Then several related methods such as HyQMOM and we refer [46,10,35] and the references therein for more details of HyQMOM and the extended methods. In [21,22], a hyperbolic quadrature based closure method is proposed by modifying the governing equations of moment coefficients at the last two orders. However, these closure methods rely on mathematical derivations, such as the Boltzmann equation and the distribution function itself.…”

Section: Introductionmentioning

confidence: 99%

Learning invariance preserving moment closure model for Boltzmann-BGK equation

Li¹,

Dong²,

Wang³

2021

Preprint

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As one of the main governing equations in kinetic theory, the Boltzmann equation is widely utilized in aerospace, microscopic flow, etc. Its high-resolution simulation is crucial in these related areas. However, due to the Boltzmann equation's high dimensionality, highresolution simulations are often difficult to achieve numerically. The moment method which Grad first proposed in 1949 [12] is among popular numerical methods to achieve efficient high-resolution simulations. We can derive the governing equations in the moment method by taking moments on both sides of the Boltzmann equation, which effectively reduces the dimensionality of the problem. However, one of the main challenges is that it leads to an unclosed moment system, and closure is needed to obtain a closed moment system. It is truly an art in designing closures for moment systems and has been a significant research field in kinetic theory. Other than the traditional human designs of closures, the machine learningbased approach has attracted much attention lately [13,19]. In this work, we propose a machine learning-based method to derive a moment closure model for the Boltzmann-BGK equation. In particular, the closure relation is approximated by a carefully designed deep neural network that possesses desirable physical invariances, i.e., the Galilean invariance, reflecting invariance, and scaling invariance, inherited from the original Boltzmann-BGK equation. Numerical simulations on the smooth and discontinuous initial condition problem, Sod shock tube problem, and the shock structure problems demonstrate satisfactory numerical performances of the proposed invariance preserving neural closure method.

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Hyperbolic moment equations using quadrature-based projection methods

Cited by 16 publications

References 7 publications

Analysis and Numerical Simulation of Hyperbolic Shallow Water Moment Equations

Analysis and Numerical Simulation of Hyperbolic Shallow Water Moment Equations

Projective integration schemes for hyperbolic moment equations

Learning invariance preserving moment closure model for Boltzmann-BGK equation

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