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Kinetic & Related Models

2014 , Volume 7 , Issue 3

Issue on traffic modeling and management: Trends and perspectives

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On hyperbolicity of 13-moment system
Zhenning Cai , Yuwei Fan and  Ruo Li
2014, 7(3): 415-432 doi: 10.3934/krm.2014.7.415 +[Abstract](35) +[PDF](1145.9KB)
We point out that the thermodynamic equilibrium is not an interior point of the hyperbolicity region of Grad's 13-moment system. With a compact expansion of the phase density, which is more compact than Grad's expansion, we derive a modified 13-moment system. The new 13-moment system admits the thermodynamic equilibrium as an interior point of its hyperbolicity region. We deduce a concise criterion to ensure the hyperbolicity, and thus the hyperbolicity region can be quantitatively depicted.
New insights into the numerical solution of the Boltzmann transport equation for photons
Taposh Kumar Das and  Óscar López Pouso
2014, 7(3): 433-461 doi: 10.3934/krm.2014.7.433 +[Abstract](36) +[PDF](1367.6KB)
This paper has been thought to describe a numerical algorithm, based on the expansion in orders of scattering, for solving the steady Boltzmann transport equation for photons, and to show some subsequent numerical results which have been obtained by developing an own Matlab® code. The spatial domain is assumed to be a rectangular-shaped container with air and a water phantom inside, albeit the method can be extended to arbitrarily shaped convex domains filled with some other heterogeneous material. High energy x-rays enter the domain through a small rectangle located on its upper face.
Numerical simulations of degenerate transport problems
Florian De Vuyst and  Francesco Salvarani
2014, 7(3): 463-476 doi: 10.3934/krm.2014.7.463 +[Abstract](47) +[PDF](575.3KB)
We consider in this article the monokinetic linear Boltzmann equation in two space dimensions with degenerate cross section and produce, by means of a finite-volume method, numerical simulations of the large-time asymptotics of the solution.
    The numerical computations are performed in the $2Dx-1Dv$ phase space on Cartesian grids and deal with both cross sections satisfying the geometrical condition and cross sections that do not satisfy it.
    The numerical simulations confirm the theoretical results on the long-time behaviour of degenerate kinetic equations for cross sections satisfying the geometrical condition. Moreover, they suggest that, for general non-trivial degenerate cross sections whose support contains a ball, the theoretical upper bound of order $t^{-1/2}$ for the time decay rate (in $L^2$-sense) can actually be reached.
A kinetic approach to error estimate for nonautonomous anisotropic degenerate parabolic-hyperbolic equations
Xingwen Hao , Yachun Li and  Qin Wang
2014, 7(3): 477-492 doi: 10.3934/krm.2014.7.477 +[Abstract](29) +[PDF](384.6KB)
This paper is devoted to the error estimate of approximate solutions for non-autonomous degenerate parabolic-hyperbolic equations, where the nonlinear convection flux, the diffusion matrix and source term depend on time $t$ explicitly. Our method is based on kinetic formulation and kinetic entropy formula. By developing the kinetic techniques, we obtain an error estimate of order $O(\sqrt{\mu})$, where $\mu$ is the artificial viscosity.
Hysteretic behavior of a moment-closure approximation for FENE model
YunKyong Hyon
2014, 7(3): 493-507 doi: 10.3934/krm.2014.7.493 +[Abstract](25) +[PDF](1292.1KB)
We discuss hysteretic behaviors of dilute viscoelastic polymeric fluids with moment-closure approximation approach in extensional/enlongational flows. Polymeric fluids are modeled by the finite extensible nonlinear elastic (FENE) spring dumbbell model. Hysteresis is one of key features to describe FENE model. We here investigate the hysteretic behavior of FENE-D model introduced in [Y. Hyon et al., Multiscale Model. Simul., 7(2008), pp.978--1002]. The FENE-D model is established from a special equilibrium solution of the Fokker-Planck equation to catch extreme behavior of FENE model in large extensional flow rates. Since the hysteresis of FENE model can be seen during a relaxation in simple extensional flow employing the normal stress/the elongational viscosity versus the mean-square extension, we simulate FENE-D in simple extensional flows to investigate its hysteretic behavior comparing to FENE-P, FENE-L [G. Lielens et al., J. Non-Newtonian Fluid Mech., 76(1999), pp.249--279]. The FENE-P is a well-known pre-averaged approximated model, and it shows a good agreement to macroscopic induced stresses. However, FENE-P does not catch any hysteretic phenomenon. In contrast, the FENE-L shows a better hysteretic behavior than the other models to FENE, but it has a limitation for macroscopic induced stresses in large shear rates. On the other hand, FENE-D presents a good agreement to macroscopic induced stresses even in large shear rates, and moreover, it shows a hysteretic phenomenon in certain large flow rates.
Approximate models for stochastic dynamic systems with velocities on the sphere and associated Fokker--Planck equations
Axel Klar , Florian Schneider and  Oliver Tse
2014, 7(3): 509-529 doi: 10.3934/krm.2014.7.509 +[Abstract](44) +[PDF](759.4KB)
We consider stochastic dynamic systems with state space $\mathbb{R}^n \times \mathbb{S}^{n-1}$ and associated Fokker--Planck equations. Such systems are used to model, for example, fiber dynamics or swarming and pedestrian dynamics with constant individual speed of propagation. Approximate equations, like linear and nonlinear (maximum entropy) moment approximations and linear and nonlinear diffusion approximations are investigated. These approximations are compared to the underlying Fokker--Planck equation with respect to quality measures like the decay rates to equilibrium. The results clearly show the superiority of the maximum entropy approach for this application compared to the simpler linear and diffusion approximations.
A framework for hyperbolic approximation of kinetic equations using quadrature-based projection methods
Julian Koellermeier , Roman Pascal Schaerer and  Manuel Torrilhon
2014, 7(3): 531-549 doi: 10.3934/krm.2014.7.531 +[Abstract](37) +[PDF](3630.5KB)
We derive hyperbolic PDE systems for the solution of the Boltzmann Equation. First, the velocity is transformed in a non-linear way to obtain a Lagrangian velocity phase space description that allows for physical adaptivity. The unknown distribution function is then approximated by a series of basis functions.
    Standard continuous projection methods for this approach yield PDE systems for the basis coefficients that are in general not hyperbolic. To overcome this problem, we apply quadrature-based projection methods which modify the structure of the system in the desired way so that we end up with a hyperbolic system of equations.
    With the help of a new abstract framework, we derive conditions such that the emerging system is hyperbolic and give a proof of hyperbolicity for Hermite ansatz functions in one dimension together with Gauss-Hermite quadrature.
One-species Vlasov-Poisson-Landau system near Maxwellians in the whole space
Yuanjie Lei , Linjie Xiong and  Huijiang Zhao
2014, 7(3): 551-590 doi: 10.3934/krm.2014.7.551 +[Abstract](23) +[PDF](674.1KB)
The classical one-species Vlasov-Poisson-Landau system describes dynamics of electrons interacting with its self-consistent electrostatic field as well as its grazing collisions modeled by the famous Landau (Fokker-Planck) collision kernel. We show in this manuscript that the Cauchy problem for the one-species Vlasov-Poisson-Landau system which includes the Coulomb potential admits a unique global solution near a given global Maxwellian in the whole space $\mathbb{R}^3_x$ provided that the initial perturbation satisfies certain regularity and smallness conditions. Compared with that of [12], which, to the best of our knowledge, is the only result concerning the one-species Vlasov-Poisson-Landau system available up to now, we do not ask the initial perturbation to satisfy the neutral condition and the minimal regularity assumption we imposed on the initial perturbation is also weaker.
On a regularized system of self-gravitating particles
René Pinnau and  Oliver Tse
2014, 7(3): 591-604 doi: 10.3934/krm.2014.7.591 +[Abstract](29) +[PDF](1259.0KB)
We consider a regularized macroscopic model describing a system of self-gravitating particles. We study the existence and uniqueness of nonnegative stationary solutions and allude the differences to results obtained from classical gravitational models. The system is analyzed on a convex, bounded domain up to three spatial dimensions, subject to Neumann boundary conditions for the particle density, and Dirichlet boundary condition for the self-interacting potential. Finally, we show numerical simulations underlining our analytical results.

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