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Kinetic & Related Models

February 2019 , Volume 12 , Issue 1

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Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks
Alin Pogan and Kevin Zumbrun
2019, 12(1): 1-36 doi: 10.3934/krm.2019001 +[Abstract](408) +[HTML](188) +[PDF](632.1KB)
Abstract:

We construct stable manifolds for a class of singular evolution equations including the steady Boltzmann equation, establishing in the process exponential decay of associated kinetic shock and boundary layers to their limiting equilibrium states. Our analysis is from a classical dynamical systems point of view, but with a number of interesting modifications to accomodate ill-posedness with respect to the Cauchy problem of the underlying evolution equation.

Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density
Fei Chen, Boling Guo and Xiaoping Zhai
2019, 12(1): 37-58 doi: 10.3934/krm.2019002 +[Abstract](381) +[HTML](170) +[PDF](487.82KB)
Abstract:

In this paper, we consider the Cauchy problem of the incompressible MHD system with discontinuous initial density in ${\mathbb R}^3$. We establish the global well-posedness of the MHD system if the initial data satisfies \begin{document}$(ρ_0, u_0, H_0)∈ L^{∞}({\mathbb R}^3)× H^s({\mathbb R}^3)× H^s({\mathbb R}^3)$\end{document} with \begin{document}$\frac{1}{2} < s \le 1$\end{document} and

for some small \begin{document}$c>0$\end{document} which only depends on \begin{document}$\underline{ρ}, \overline{ρ}$\end{document}. As a byproduct, we also get the decay estimate of the solution.

A stochastic algorithm without time discretization error for the Wigner equation
Orazio Muscato and Wolfgang Wagner
2019, 12(1): 59-77 doi: 10.3934/krm.2019003 +[Abstract](395) +[HTML](163) +[PDF](551.38KB)
Abstract:

Stochastic particle methods for the numerical treatment of the Wigner equation are considered. The approximation properties of these methods depend on several numerical parameters. Such parameters are the number of particles, a time step (if transport and other processes are treated separately) and the grid size (used for the discretization of the position and the wave-vector). A stochastic algorithm without time discretization error is introduced. Its derivation is based on the theory of piecewise deterministic Markov processes. Numerical experiments are performed in a one-dimensional test case. Approximation properties with respect to the grid size and the number of particles are studied. Convergence of a time-splitting scheme to the no-splitting algorithm is demonstrated. The no-splitting algorithm is shown to be more efficient in terms of computational effort.

Numerical solutions for multidimensional fragmentation problems using finite volume methods
Jitraj Saha, Nilima Das, Jitendra Kumar and Andreas Bück
2019, 12(1): 79-103 doi: 10.3934/krm.2019004 +[Abstract](650) +[HTML](143) +[PDF](531.16KB)
Abstract:

We introduce a finite volume scheme for approximating a general multidimensional fragmentation problem. The scheme estimates several physically significant moment functions with good accuracy, and is robust with respect to use of different nonuniform daughter distribution functions. Moreover, it possess simple mathematical formulation for defining in higher dimensions. The efficiency of the scheme is validated over several test problems.

On hp-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system
Mohammad Asadzadeh, Piotr Kowalczyk and Christoffer Standar
2019, 12(1): 105-131 doi: 10.3934/krm.2019005 +[Abstract](314) +[HTML](137) +[PDF](770.63KB)
Abstract:

We study stability and convergence of \begin{document}$hp$\end{document}-streamline diffusion (SD) finite element, and Nitsche's schemes for the three dimensional, relativistic (3 spatial dimension and 3 velocities), time dependent Vlasov-Maxwell system and Maxwell's equations, respectively. For the \begin{document}$hp$\end{document} scheme for the Vlasov-Maxwell system, assuming that the exact solution is in the Sobolev space \begin{document}$H^{s+1}(Ω)$\end{document}, we derive global a priori error bound of order \begin{document}${\mathcal O}(h/p)^{s+1/2}$\end{document}, where \begin{document}$h ( = \max_K h_K)$\end{document} is the mesh parameter and \begin{document}$p ( = \max_K p_K)$\end{document} is the spectral order. This estimate is based on the local version with \begin{document}$h_K = \mbox{ diam } K$\end{document} being the diameter of the phase-space-time element \begin{document}$K$\end{document} and \begin{document}$p_K$\end{document} is the spectral order (the degree of approximating finite element polynomial) for \begin{document}$K$\end{document}. As for the Nitsche's scheme, by a simple calculus of the field equations, first we convert the Maxwell's system to an elliptic type equation. Then, combining the Nitsche's method for the spatial discretization with a second order time scheme, we obtain optimal convergence of \begin{document}${\mathcal O}(h^2+k^2)$\end{document}, where \begin{document}$h$\end{document} is the spatial mesh size and \begin{document}$k$\end{document} is the time step. Here, as in the classical literature, the second order time scheme requires higher order regularity assumptions. Numerical justification of the results, in lower dimensions, is presented and is also the subject of a forthcoming computational work [22].

Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations
Roberta Bianchini and Roberto Natalini
2019, 12(1): 133-158 doi: 10.3934/krm.2019006 +[Abstract](297) +[HTML](153) +[PDF](381.73KB)
Abstract:

We present a rigorous convergence result for smooth solutions to a singular semilinear hyperbolic approximation, called vector-BGK model, to the solutions to the incompressible Navier-Stokes equations in Sobolev spaces. Our proof deeply relies on the dissipative properties of the system and on the use of an energy which is provided by a symmetrizer, whose entries are weighted in a suitable way with respect to the singular perturbation parameter. This strategy allows us to perform uniform energy estimates and to prove the convergence by compactness.

Elastic limit and vanishing external force for granular systems
Fei Meng and Xiao-Ping Yang
2019, 12(1): 159-176 doi: 10.3934/krm.2019007 +[Abstract](396) +[HTML](149) +[PDF](378.73KB)
Abstract:

We consider two popular models derived from the theory of granular gases. The first model is the inelastic Boltzmann equation with a diffusion term representing the heat bath, the second model is obtained by a self-similar transformation for the inelastic Boltzmann equation in the homogeneous cooling problem. We prove that the steady states of the two models converge to a Maxwellian equilibrium or a Dirac distribution in the elastic limit and the vanishing external force, respectively. Our results show that the limits of the steady states depend on the ratio of external energy and dissipated energy due to inelastic collision. These results provide a partial answer to a question proposed by Gamba, Panferov and Villani (Comm. Math. Phys. 246,503-541. 2004).

Kinetic models and intrinsic timescales: Simulation comparison for a 2nd order queueing model
Dieter Armbruster and Matthew Wienke
2019, 12(1): 177-193 doi: 10.3934/krm.2019008 +[Abstract](464) +[HTML](168) +[PDF](730.4KB)
Abstract:

Kinetic models of stochastic production flows can be expanded into deterministic moment equations and thus approximated with appropriate closures. A second order model for the product density and the product speed has previously been proposed. A systematic analysis comparing simulations of the partial differential equations (PDE) with discrete event simulations (DES) is performed. Specifically, factory production is modeled as an M/M/1 queue where the arrival process is a non-homogeneous Poisson process. Three fundamental scenarios for such a time dependent influx are studied: An instant step up/step down of the arrival rate, an exponential step up/step down and periodic variation of the average arrival rate. It is shown that the second order model in general yields significant improvements over the first order model. Adding diffusion into the PDE further improves the agreement in particular for queues with low utilization. The analysis also points to fundamental open issues regarding kinetic models of time dependent agent based simulations. Memory effects and the possibility of resonance in deterministic models are caused by intrinsic timescales of the PDE that are not present in the original stochastic processes.

An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions
Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová and Șeyma Nur Özcan
2019, 12(1): 195-216 doi: 10.3934/krm.2019009 +[Abstract](428) +[HTML](200) +[PDF](2327.4KB)
Abstract:

In this paper, we study two-dimensional multiscale chemotaxis models based on a combination of the macroscopic evolution equation for chemoattractant and microscopic models for cell evolution. The latter is governed by a Boltzmann-type kinetic equation with a local turning kernel operator which describes the velocity change of the cells. The parabolic scaling yields a non-dimensional kinetic model with a small parameter, which represents the mean free path of the cells. We propose a new asymptotic preserving numerical scheme that reflects the convergence of the studied micro-macro model to its macroscopic counterpart-the Patlak-Keller-Segel system-in the singular limit. The method is based on the operator splitting strategy and a suitable combination of the higher-order implicit and explicit time discretizations. In particular, we use the so-called even-odd decoupling and approximate the stiff terms arising in the singular limit implicitly. We prove that the resulting scheme satisfies the asymptotic preserving property. More precisely, it yields a consistent approximation of the Patlak-Keller-Segel system as the mean-free path tends to 0. The derived asymptotic preserving method is used to get better insight to the blowup behavior of two-dimensional kinetic chemotaxis model.

A quantum Drift-Diffusion model and its use into a hybrid strategy for strongly confined nanostructures
Clément Jourdana and Paola Pietra
2019, 12(1): 217-242 doi: 10.3934/krm.2019010 +[Abstract](446) +[HTML](191) +[PDF](1843.94KB)
Abstract:

In this paper we derive by an entropy minimization technique a local Quantum Drift-Diffusion (QDD) model that allows to describe with accuracy the transport of electrons in confined nanostructures. The starting point is an effective mass model, obtained by considering the crystal lattice as periodic only in the one dimensional longitudinal direction and keeping an atomistic description of the entire two dimensional cross-section. It consists of a sequence of one dimensional device dependent Schrödinger equations, one for each energy band, in which quantities retaining the effects of the confinement and of the transversal crystal structure are inserted. These quantities are incorporated into the definition of the entropy and consequently the QDD model that we obtain has a peculiar quantum correction that includes the contributions of the different energy bands. Next, in order to simulate the electron transport in a gate-all-around Carbon Nanotube Field Effect Transistor, we propose a spatial hybrid strategy coupling the QDD model in the Source/Drain regions and the Schrödinger equations in the channel. Self-consistent computations are performed coupling the hybrid transport equations with the resolution of a Poisson equation in the whole three dimensional domain.

Time-splitting methods to solve the Hall-MHD systems with Lévy noises
Zhong Tan, Huaqiao Wang and Yucong Wang
2019, 12(1): 243-267 doi: 10.3934/krm.2019011 +[Abstract](321) +[HTML](155) +[PDF](491.35KB)
Abstract:

In this paper, we establish the existence of a martingale solution to the stochastic incompressible Hall-MHD systems with Lévy noises in a bounded domain. The proof is based on a new method, i.e., the time splitting method and the stochastic compactness method.

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