This paper presents second-order accurate genuine BGK (Bhatnagar-Gross-Krook) schemes in the framework of finite volume method for the ultra-relativistic flows. Different from the existing kinetic flux-vector splitting (KFVS) or BGK-type schemes for the ultra-relativistic Euler equations, the present genuine BGK schemes are derived from the analytical solution of the Anderson-Witting model, which is given for the first time and includes the "genuine" particle collisions in the gas transport process. The BGK schemes for the ultra-relativistic viscous flows are also developed and two examples of ultra-relativistic viscous flow are designed. Several 1D and 2D numerical experiments are conducted to demonstrate that the proposed BGK schemes not only are accurate and stable in simulating ultra-relativistic inviscid and viscous flows, but also have higher resolution at the contact discontinuity than the KFVS or BGK-type schemes.

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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 the complicate Grad type orthogonal polynomials depending on a parameter and the real spherical harmonics. We derive the recurrence relations of the polynomials, calculate their derivatives with respect to the independent variable and parameter respectively, and study their zeros. The recurrence relations and partial derivatives of the real spherical harmonics are also given. It is proved that our moment system is globally hyperbolic, and linearly stable. Moreover, the Lorentz covariance is also studied in the 1D space.

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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