Multiscale Schemes for the BGK--Vlasov--Poisson System in the Quasi-Neutral and Fluid Limits. Stability Analysis and First Order Schemes

This paper deals with the development and the analysis of asymptotically stable and consistent schemes in the joint quasi-neutral and fluid limits for the collisional Vlasov-Poisson system. In these limits, the classical explicit schemes suffer from time step restrictions due to the small plasma period and Knudsen number. To solve this problem, we propose a new scheme stable for choices of time steps independent from the small scales dynamics and with comparable computational cost with respect to standard explicit schemes. In addition, this scheme reduces automatically to consistent discretizations of the underlying asymptotic systems. In this first work on this subject, we propose a first order in time scheme and we perform a relative linear stability analysis to deal with such problems. The framework we propose permits to extend this approach to high order schemes in the next future. We finally show the capability of the method in dealing with small scales through numerical experiments.

Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system

We construct a new family of semi-discrete numerical schemes for the approximation of the one-dimensional periodic Vlasov-Poisson system. The methods are based on the coupling of discontinuous Galerkin approximation to the Vlasov equation and several finite element (conforming, non-conforming and mixed) approximations for the Poisson problem. We show optimal error estimates for the all proposed methods in the case of smooth compactly supported initial data. The issue of energy conservation is also analyzed for some of the methods.

An energy preserving discontinouous Galerkin method for Vlasov-Poisson system

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Discontinuous Galerkin methods for the Multi-dimensional Vlasov-Poisson problem

We introduce and analyze two new semi-discrete numerical methods for the multi-dimensional Vlasov-Poisson system. The schemes are constructed by combing a discontinuous Galerkin approximation to the Vlasov equation together with a mixed finite element method for the Poisson problem. We show optimal error estimates in the case of smooth compactly supported initial data. We propose a scheme that preserves the total energy of the system.

Asymptotic Analysis of a Schrödinger-Poisson System with Quantum Wells and Macroscopic Nonlinearities in Dimension 1

2000 Mathematics Subject Classification: 35Q02, 35Q05, 35Q10, 35B40.

Relativistic current collection by a cylindrical Langmuir probe

The current I to a cylindrical Langmuir probe with a bias Φp satisfying β≡eΦp/mec2∼O(1) is discussed. The probe is considered at rest in an unmagnetized plasma composed of electrons and ions with temperatureskTe∼kTi≪mec2. For small enough radius, the probe collects the relativistic orbital-motion-limited (OML) current I OML , which is shown to be larger than the non-relativistic result; the OML current is proportional to β1/2 and β3/2 in the limits β≪1 and β≫1, respectively. Unlike the non-relativistic case, the electron density can exceed the unperturbed density value. An asymptotic theory allowed to compute the maximum radius of the probe to collect OML current, the sheath radius for probe radius well below maximum and how the ratio I/I OML drops below unity when the maximum radius is exceeded. A numerical algorithm that solves the Vlasov-Poisson system was implemented and density and potential profiles presented. The results and their implications in a possible mission to Jupiter with electrodynamic bare tethers are discussed density value. An asymptotic theory allowed to compute the maximum radius of the probe to collect OML current, the sheath radius for probe radius well below maximum and how the ratio I/IOML drops below unity when the maximum radius is exceeded. A numerical algorithm that solves the Vlasov-Poisson system was implemented and density and potential profiles presented. The results and their implications in a possible mission to Jupiter with electrodynamic bare tethers are discussed.

Existence of solutions and asymtptotic limits of the Euler-Poisson and the quantum drift-diffusion systems

My work concerns two different systems of equations used in the mathematical modeling of semiconductors and plasmas: the Euler-Poisson system and the quantum drift-diffusion system. The first is given by the Euler equations for the conservation of mass and momentum, with a Poisson equation for the electrostatic potential. The second one takes into account the physical effects due to the smallness of the devices (quantum effects). It is a simple extension of the classical drift-diffusion model which consists of two continuity equations for the charge densities, with a Poisson equation for the electrostatic potential. Using an asymptotic expansion method, we study (in the steady-state case for a potential flow) the limit to zero of the three physical parameters which arise in the Euler-Poisson system: the electron mass, the relaxation time and the Debye length. For each limit, we prove the existence and uniqueness of profiles to the asymptotic expansion and some error estimates. For a vanishing electron mass or a vanishing relaxation time, this method gives us a new approach in the convergence of the Euler-Poisson system to the incompressible Euler equations. For a vanishing Debye length (also called quasineutral limit), we obtain a new approach in the existence of solutions when boundary layers can appear (i.e. when no compatibility condition is assumed). Moreover, using an iterative method, and a finite volume scheme or a penalized mixed finite volume scheme, we numerically show the smallness condition on the electron mass needed in the existence of solutions to the system, condition which has already been shown in the literature. In the quantum drift-diffusion model for the transient bipolar case in one-space dimension, we show, by using a time discretization and energy estimates, the existence of solutions (for a general doping profile). We also prove rigorously the quasineutral limit (for a vanishing doping profile). Finally, using a new time discretization and an algorithmic construction of entropies, we prove some regularity properties for the solutions of the equation obtained in the quasineutral limit (for a vanishing pressure). This new regularity permits us to prove the positivity of solutions to this equation for at least times large enough.

Direct Vlasov simulations of electron-attracting cylindrical Langmuir probes in flowing plasmas

Current collection by positively polarized cylindrical Langmuir probes immersed in flowing plasmas is analyzed using a non-stationary direct Vlasov-Poisson code. A detailed description of plasma density spatial structure as a function of the probe-to-plasma relative velocity U is presented. Within the considered parametric domain, the well-known electron density maximum close to the probe is weakly affected by U. However, in the probe wake side, the electron density minimum becomes deeper as U increases and a rarified plasma region appears. Sheath radius is larger at the wake than at the front side. Electron and ion distribution functions show specific features that are the signature of probe motion. In particular, the ion distribution function at the probe front side exhibits a filament with positive radial velocity. It corresponds to a population of rammed ions that were reflected by the electric field close to the positively biased probe. Numerical simulations reveal that two populations of trapped electrons exist: one orbiting around the probe and the other with trajectories confined at the probe front side. The latter helps to neutralize the reflected ions, thus explaining a paradox in past probe theory.

A Kinetic Vlasov Model for Plasma Simulation Using Discontinuous Galerkin Method on Many-Core Architectures

Thesis (Ph.D.)--University of Washington, 2016-03

Solution of the Poisson-Boltzmann equation for a system with four ionic species

The analytical solution of the Poisson-Boltzmann equation in an electrolyte with four ionic species (2:2:1:1), in the presence of a charged planar membrane or surface is presented. The function describing the mean electrical potential provides a convenient description that helps the understanding of electrical processes of biological interest.

Insights into collisional magmatism from isotopic fingerprints of melting reactions

Piston-cylinder experiments in the granite system demonstrate that a variety of isotopically distinct melts can arise from progressive melting of a single source. The relation between the isotopic composition of Sr and the stoichiometry of the observed melting reactions suggests that isotopic signatures of anatectic magmas can be used to infer melting reactions in natural systems. Our results also indicate that distinct episodes of dehydration and fluid-fluxed melting of a single, metapelitic source region may have contributed to the bimodal geochemistry of crustally derived leucogranites of the Himalayan orogen.

Optimal designs for mixed effects poisson regression models

Magdeburg, Univ., Fak. für Mathematik, Diss., 2010

A priori ratemaking using bivariate poisson regression models

In automobile insurance, it is useful to achieve a priori ratemaking by resorting to gene- ralized linear models, and here the Poisson regression model constitutes the most widely accepted basis. However, insurance companies distinguish between claims with or without bodily injuries, or claims with full or partial liability of the insured driver. This paper exa- mines an a priori ratemaking procedure when including two di®erent types of claim. When assuming independence between claim types, the premium can be obtained by summing the premiums for each type of guarantee and is dependent on the rating factors chosen. If the independence assumption is relaxed, then it is unclear as to how the tari® system might be a®ected. In order to answer this question, bivariate Poisson regression models, suitable for paired count data exhibiting correlation, are introduced. It is shown that the usual independence assumption is unrealistic here. These models are applied to an automobile insurance claims database containing 80,994 contracts belonging to a Spanish insurance company. Finally, the consequences for pure and loaded premiums when the independence assumption is relaxed by using a bivariate Poisson regression model are analysed.

Irreversible thermodynamics of Poisson processes with reaction

A kinetic model is derived to study the successive movements of particles, described by a Poisson process, as well as their generation. The irreversible thermodynamics of this system is also studied from the kinetic model. This makes it possible to evaluate the differences between thermodynamical quantities computed exactly and up to second-order. Such differences determine the range of validity of the second-order approximation to extended irreversible thermodynamics

Rate of convergence of a particle method to the solution of the Mc Kean-Vlasov's equation

This paper studies the rate of convergence of an appropriatediscretization scheme of the solution of the Mc Kean-Vlasovequation introduced by Bossy and Talay. More specifically,we consider approximations of the distribution and of thedensity of the solution of the stochastic differentialequation associated to the Mc Kean - Vlasov equation. Thescheme adopted here is a mixed one: Euler/weakly interactingparticle system. If $n$ is the number of weakly interactingparticles and $h$ is the uniform step in the timediscretization, we prove that the rate of convergence of thedistribution functions of the approximating sequence in the $L^1(\Omega\times \Bbb R)$ norm and in the sup norm is of theorder of $\frac 1{\sqrt n} + h $, while for the densities is ofthe order $ h +\frac 1 {\sqrt {nh}}$. This result is obtainedby carefully employing techniques of Malliavin Calculus.