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This paper extends the model reduction method by the operator projection to the one-dimensional special relativistic Boltzmann equation. The derivation of arbitrary order globally hyperbolic moment system is built on our careful study of…
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…
We make a brief historical review to the moment model reduction to the kinetic equations, particularly the Grad's moment method for Boltzmann equation. The focus is on the hyperbolicity of the reduced model, which is essential to the…
This paper extends the model reduction method by the operator projection to the three-dimensional special relativistic Boltzmann equation. The derivation of arbitrary order moment system is built on our careful study of infinite families of…
In this paper, we study a nonlinear system of first order partial differential equations describing the macroscopic behavior of an ensemble of interacting self-propelled rigid bodies. Such system may be relevant for the modelling of bird…
We consider the dynamics of the system of self propelling particles modeled via the Vicsek algorithm in continuum time limit. It is shown that the alignment process for the velocities can be subdivided into two regimes: "fast" kinetic and…
The Vicsek model is a very popular individual based model which describes collective behavior among animal societies. A macroscopic version of the Vicsek model has been derived from a large scale limit of this individual based model…
We introduce a numerical method for solving Grad's moment equations or regularized moment equations for arbitrary order of moments. In our algorithm, we do not need explicitly the moment equations. As an instead, we directly start from the…
We model the Knudsen layer in Kramers' problem by linearized high order hyperbolic moment system. Due to the hyperbolicity, the boundary conditions of the moment system is properly reduced from the kinetic boundary condition. For Kramers'…
In this paper, we propose a globally hyperbolic regularization to the general Grad's moment system in multi-dimensional spaces. Systems with moments up to an arbitrary order are studied. The characteristic speeds of the regularized moment…
We introduce a hyperbolic closure for the Grad moment expansion of the Bhatnagar-Gross-Krook's (BGK) kinetic model using a neural network (NN) trained on BGK's moment data. This closure is motivated by the exact closure for the free…
We study dynamic self-organisation and order-disorder transitions in a two-dimensional system of self-propelled particles. Our model is a variation of the Vicsek model, where particles align the motion to their neighbours but repel each…
In this paper, we present a regularization to 1D Grad's moment system to achieve global hyperbolicity. The regularization is based on the observation that the characteristic polynomial of the Jacobian of the flux in Grad's moment system is…
By a further investigation on the structure of the coefficient matrix of the globally hyperbolic regularized moment equations for Boltzmann equation in [Z. Cai, Y. Fan and R. Li, Comm. Math. Sci., 11 (2013), pp. 547-571], we propose a…
Geophysical flow simulations using hyperbolic shallow water moment equations require an efficient discretization of a potentially large system of PDEs, the so-called moment system. This calls for tailored model order reduction techniques…
In this paper, we derive the quantum hydrodynamics models based on the moment closure of the Wigner equation. The moment expansion adopted is of the Grad type firstly proposed in \cite{Grad}. The Grad's moment method was originally…
We present a numerical discretisation of the coupled moment systems, previously introduced in Dahm and Helzel, which approximate the kinetic multi-scale model by Helzel and Tzavaras for sedimentation in suspensions of rod-like particles for…
To close the moment model deduced from kinetic equations, the canonical approach is to provide an approximation to the flux function not able to be depicted by the moments in the reduced model. In this paper, we propose a brand new closure…
We present a general, high-order, fully explicit relaxation scheme which can be applied to any system of nonlinear hyperbolic conservation laws in multiple dimensions. The scheme consists of two steps. In a first (relaxation) step, the…
Kinetic simulations are computationally intensive due to six-dimensional phase space discretization. Many kinetic spectral solvers use the asymmetrically weighted Hermite expansion due to its conservation and fluid-kinetic coupling…