A framework for hyperbolic approximation of kinetic equations using quadrature-based projection methods

Koellermeier, Julian; Schaerer, Roman Pascal; Torrilhon, Manuel

doi:10.3934/krm.2014.7.531

Cited by 57 publications

(64 citation statements)

References 19 publications

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“…are the quadrature-based moment equations [12], and the same as those in Section 5.5. From the point of view of using different projections in the framework, we can see the difference between HME and QBME for 1D, which is only the use of a different projection operator.…”

Section: If We Choose Pmentioning

confidence: 99%

Model Reduction of Kinetic Equations by Operator Projection

et al. 2015

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By a further study of the mechanism of the hyperbolic regularization of the moment system for Boltzmann equation proposed in [Z. Cai, Y. Fan, R. Li, Comm. Math. Sci. 11(2): 547-571, 2013], we point out that the key point is treating the time and space derivative in the same way. Based on this understanding, a uniform framework to derive globally hyperbolic moment systems from kinetic equations using an operator projection method is proposed. The framework is so concise and clear that it can be treated as an algorithm with four inputs to derive hyperbolic moment system by routine calculations. Almost all existing globally hyperbolic moment system can be included in the framework, as well as some new moment system including globally hyperbolic regularized versions of Grad ordered moment system and a multidimensional extension of the quadrature-based moment system.Comment: 32 pages, 2 figure

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Section: If We Choose Pmentioning

confidence: 99%

Model Reduction of Kinetic Equations by Operator Projection

et al. 2015

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“…This method was extended to the multi-dimensional case in two ways [7,22]. Shortly after that, in [30], a quadrature-based moment method was proposed by computing the integrals using a suitable quadrature rule instead of exact integration. This method also yields globally hyperbolic moment equations, called Quadrature-Based Moment Equations (QBME).…”

Section: Introductionmentioning

confidence: 99%

“…In recent years, the hyperbolic regularization in [6,21] has been applied to a lot of fields besides gas kinetic theory and microflow, including semiconductor device simulation [10], plasma simulation [14,19], density functional theory [11], quantum gas kinetic theory [17] and rarefied relativistic Boltzmann equation [36]. However, the complexity of the regularized moment models limited their further application and the theoretical and numerical comparison between the regularizations in [6,7] and [30] is not rich. In particular the derivation gives little insight into the explicit form of the equations.…”

Section: Introductionmentioning

confidence: 99%

Diagram notation for the derivation of hyperbolic moment systems

Koellermeier

Fan

2020

Communications in Mathematical Sciences

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We propose a diagram notation for the derivation of hyperbolic moment models for the Boltzmann equation that yields a better understanding of the resulting moment systems. So far several hyperbolic moment models were presented, but their derivations are often very technical and there is little insight into the explicit form of the equations. In our diagram notation, each term in the moment equations can be explicitly tracked throughout the derivation process and whether the resulting moment system is hyperbolic can be easily observed from the diagram. We apply the diagram notation to derive existing moment models, including Grad's moment equations, hyperbolic moment equations, quadrature-based moment equations, and a new set of simplified hyperbolic moment equations that was rarely studied before. The differences in the derivation are easily explained in the diagram notation and the explicit form of the equations can be computed straightforwardly as opposed to existing frameworks.

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“…Based on the observation that the characteristic polynomial of the flux Jacobian in the Grad moment system did not depend on the intermediate moments, a regularization was presented in [2,3,4] for the one-and multi-dimensional Grad moment systems to achieve global hyperbolicity. The quadrature based projection methods were used to derive hyperbolic systems for the solution of the Boltzmann equation [18,19] by using the quadrature rule instead of the exact integration. In the 1D case, it is similar to the regularization in [2].…”

Section: Introductionmentioning

confidence: 99%

Model Reduction of a Two-Dimensional Kinetic Swarming Model by Operator Projections

Duan¹,

Kuang²,

Tang³

2018

EAJAM

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This paper derives the arbitrary order globally hyperbolic moment system for a non-linear kinetic description of the Vicsek swarming model by using the operator projection. It is built on our careful study of a family of the complicate Grad type orthogonal functions depending on a parameter (angle of macroscopic velocity). We calculate their derivatives with respect to the independent variable, and projection of those derivatives, the product of velocity and basis, and collision term. The moment system is also proved to be hyperbolic, rotational invariant, and mass-conservative. The relationship between Grad type expansions in different parameter is also established. A semi-implicit numerical scheme is presented to solve a Cauchy problem of our hyperbolic moment system in order to verify the convergence behavior of the moment method. It is also compared to the spectral method for the kinetic equation. The results show that the solutions of our hyperbolic moment system converge to the solutions of the kinetic equation for the Vicsek model as the order of the moment system increases, and the moment method can capture key features such as vortex formation and traveling waves.the notation Id is the identity operator, Ω(t, x) is the mean velocity, and J (t, x) denotes the mean flux at x and is defined by(2.3)Here K(y − x) is the characteristic function of the ball B(0, R) = {x : |x| ≤ R}, i.e. K(x) = 1 |x|

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A framework for hyperbolic approximation of kinetic equations using quadrature-based projection methods

Cited by 57 publications

References 19 publications

Model Reduction of Kinetic Equations by Operator Projection

Model Reduction of Kinetic Equations by Operator Projection

Diagram notation for the derivation of hyperbolic moment systems

Model Reduction of a Two-Dimensional Kinetic Swarming Model by Operator Projections

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