A unified gas kinetic scheme for transport and collision effects in plasma

In this study, the Vlasov-Poisson equation with or without collision term for plasma is solved by the unified gas kinetic scheme (UGKS). The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction. The distribution function is discretized in discrete particle velocity space. After the Vlasov equation is integrated in finite volumes of physical space, the numerical flux across a cell interface and source term for particle acceleration are computed to update the distribution function at next time step. The flux is decided by Riemann problem and variation of distribution function in discrete particle velocity space is evaluated with central difference method. A electron-ion collision model is introduced in the Vlasov equation. This finite volume method for the UGKS couples the free transport and long-range interaction between particles. The electric field induced by charged particles is controlled by the Poisson's equation. In this paper, the Poisson's equation is solved using the Green's function for two dimensional plasma system subjected to the symmetry or periodic boundary conditions. Two numerical tests of the linear Landau damping and the Gaussian beam are carried out to validate the proposed method. The linear electron plasma wave damping is simulated based on electron-ion collision operator. Compared with previous methods, it is shown that the current method is able to obtain accurate results of the Vlasov-Poisson equation with a time step much larger than the particle collision time. Highly non-equilibrium and rarefied plasma flows, such as electron flows driven by electromagnetic field, can be simulated easily.Comment: 33 pages, 13 figure

Estimation of the infinitesimal generator by square-root approximation

For the analysis of molecular processes, the estimation of time-scales, i.e., transition rates, is very important. Estimating the transition rates between molecular conformations is -- from a mathematical point of view -- an invariant subspace projection problem. A certain infinitesimal generator acting on function space is projected to a low-dimensional rate matrix. This projection can be performed in two steps. First, the infinitesimal generator is discretized, then the invariant subspace is approxi-mated and used for the subspace projection. In our approach, the discretization will be based on a Voronoi tessellation of the conformational space. We will show that the discretized infinitesimal generator can simply be approximated by the geometric average of the Boltzmann weights of the Voronoi cells. Thus, there is a direct correla-tion between the potential energy surface of molecular structures and the transition rates of conformational changes. We present results for a 2d-diffusion process and Alanine dipeptide

Calibration of lubrication force measurements by lattice Boltzmann simulations

This paper was presented at the 2nd Micro and Nano Flows Conference (MNF2009), which was held at Brunel University, West London, UK. The conference was organised by Brunel University and supported by the Institution of Mechanical Engineers, IPEM, the Italian Union of Thermofluid dynamics, the Process Intensification Network, HEXAG - the Heat Exchange Action Group and the Institute of Mathematics and its Applications.Many experiments explore the hydrodynamic boundary of a surface by approaching a colloidal sphere and measuring the occurring lubrication force. However, in this case many different parameters like wettability and surface roughness influence the result. In the experiment these cannot be separated easily. For a deeper understanding of such surface effects a tool is required that predicts the influence of different surface properties. Here computer simulations can help. In this paper we present lattice Boltzmann simulations of a sphere submerged in a Newtonian liquid and show that our method is able to reproduce the theoretical predictions. In order to provide high precision simulation results the influence of finite size effects has to be controlled. We study the influence of the required system size and resolution of the sphere and demonstrate that already moderate computing ressources allow to keep the error below 1%.This study is funded by DFG priority program SPP 1164

Poisson -- Boltzmann Brownian Dynamics of Charged Colloids in Suspension

We describe a method to simulate the dynamics of charged colloidal particles suspended in a liquid containing dissociated ions and salt ions. Regimes of prime current interest are those of large volume fraction of colloids, highly charged particles and low salt concentrations. A description which is tractable under these conditions is obtained by treating the small dissociated and salt ions as continuous fields, while keeping the colloidal macroions as discrete particles. For each spatial configuration of the macroions, the electrostatic potential arising from all charges in the system is determined by solving the nonlinear Poisson--Boltzmann equation. From the electrostatic potential, the forces acting on the macroions are calculated and used in a Brownian dynamics simulation to obtain the motion of the latter. The method is validated by comparison to known results in a parameter regime where the effective interaction between the macroions is of a pairwise Yukawa form

A new multidimensional, energy-dependent two-moment transport code for neutrino-hydrodynamics

We present the new code ALCAR developed to model multidimensional, multi energy-group neutrino transport in the context of supernovae and neutron-star mergers. The algorithm solves the evolution equations of the 0th- and 1st-order angular moments of the specific intensity, supplemented by an algebraic relation for the 2nd-moment tensor to close the system. The scheme takes into account frame-dependent effects of order O(v/c) as well as the most important types of neutrino interactions. The transport scheme is significantly more efficient than a multidimensional solver of the Boltzmann equation, while it is more accurate and consistent than the flux-limited diffusion method. The finite-volume discretization of the essentially hyperbolic system of moment equations employs methods well-known from hydrodynamics. For the time integration of the potentially stiff moment equations we employ a scheme in which only the local source terms are treated implicitly, while the advection terms are kept explicit, thereby allowing for an efficient computational parallelization of the algorithm. We investigate various problem setups in one and two dimensions to verify the implementation and to test the quality of the algebraic closure scheme. In our most detailed test, we compare a fully dynamic, one-dimensional core-collapse simulation with two published calculations performed with well-known Boltzmann-type neutrino-hydrodynamics codes and we find very satisfactory agreement.Comment: 30 pages, 12 figures. Revised version: several additional comments and explanations, results remain unchanged. Accepted for publication in MNRA

Solving Vlasov Equations Using NRxx Method

In this paper, we propose a moment method to numerically solve the Vlasov equations using the framework of the NRxx method developed in [6, 8, 7] for the Boltzmann equation. Due to the same convection term of the Boltzmann equation and the Vlasov equation, it is very convenient to use the moment expansion in the NRxx method to approximate the distribution function in the Vlasov equations. The moment closure recently presented in [5] is applied to achieve the globally hyperbolicity so that the local well-posedness of the moment system is attained. This makes our simulations using high order moment expansion accessible in the case of the distribution far away from the equilibrium which appears very often in the solution of the Vlasov equations. With the moment expansion of the distribution function, the acceleration in the velocity space results in an ordinary differential system of the macroscopic velocity, thus is easy to be handled. The numerical method we developed can keep both the mass and the momentum conserved. We carry out the simulations of both the Vlasov-Poisson equations and the Vlasov-Poisson-BGK equations to study the linear Landau damping. The numerical convergence is exhibited in terms of the moment number and the spatial grid size, respectively. The variation of discretized energy as well as the dependence of the recurrence time on moment order is investigated. The linear Landau damping is well captured for different wave numbers and collision frequencies. We find that the Landau damping rate linearly and monotonically converges in the spatial grid size. The results are in perfect agreement with the theoretic data in the collisionless case