Bokai Yan

A Successive Penalty-Based Asymptotic-Preserving Scheme for Kinetic Equations

We propose an asymptotic-preserving (AP) scheme for kinetic equations that is efficient also in the hydrodynamic regimes. This scheme is based on the Bhantnagar--Gross--Krook (BGK) penalty method introduced by Filbet and Jin [J. Comput. Phys., 229 (2010), pp. 7625--7648], but uses the penalization successively to achieve the desired asymptotic property. This method possesses a stronger AP property than the original method of Filbet and Jin, with the additional feature of being positivity preserving when applied on the Boltzmann equation. It is also general enough to be applicable to several important classes of kinetic equations, including the Boltzmann equation and the Landau equation. Numerical experiments verify these properties.

Asymptotic-Preserving schemes for kinetic-fluid modeling of disperse two-phase flows

Abstract We consider a system coupling the incompressible Navier–Stokes equations to the Vlasov–Fokker–Planck equation. Such a problem arises in the description of particulate flows. We design a numerical scheme to simulate the behavior of the system. This scheme is asymptotic-preserving, thus efficient in both the kinetic and hydrodynamic regimes. It has a numerical stability condition controlled by the non-stiff convection operator, with an implicit treatment of the stiff drag term and the Fokker–Planck operator. Yet, consistent to a standard asymptotic-preserving Fokker–Planck solver or an incompressible Navier–Stokes solver, only the conjugate–gradient method and fast Poisson and Helmholtz solvers are needed. Numerical experiments are presented to demonstrate the accuracy and asymptotic behavior of the scheme, with several interesting applications.

A hybrid method with deviational particles for spatial inhomogeneous plasma

In this work we propose a Hybrid method with Deviational Particles (HDP) for a plasma modeled by the inhomogeneous Vlasov-Poisson-Landau system. We split the distribution into a Maxwellian part evolved by a grid based fluid solver and a deviation part simulated by numerical particles. These particles, named deviational particles, could be both positive and negative. We combine the Monte Carlo method proposed in 31, a Particle in Cell method and a Macro-Micro decomposition method 3 to design an efficient hybrid method. Furthermore, coarse particles are employed to accelerate the simulation. A particle resampling technique on both deviational particles and coarse particles is also investigated and improved. This method is applicable in all regimes and significantly more efficient compared to a PIC-DSMC method near the fluid regime.

Analysis and simulation for a model of electron impact excitation/deexcitation and ionization/recombination

This paper describes a kinetic model and a corresponding Monte Carlo simulation method for excitation/deexcitation and ionization/recombination by electron impact in a plasma free of external fields. The atoms and ions in the plasma are represented by continuum densities and the electrons by a particle distribution. A Boltzmann-type equation is formulated and a corresponding H-theorem is formally derived. An efficient Monte Carlo method is developed for an idealized analytic model of the excitation and ionization collision cross sections. To accelerate the simulation, the reduced rejection method and binary search method are used to overcome the singular rate in the recombination process. Numerical results are presented to demonstrate the efficiency of the method on spatially homogeneous problems. The evolution of the electron distribution function and atomic states is studied, revealing the possibility under certain circumstances of system relaxation towards stationary states that are not the equilibrium states, a potential non-ergodic behavior.

A Monte Carlo method with negative particles for Coulomb collisions

In this work we propose a novel negative particle method for the general bilinear collision operators in the spatial homogeneous case and apply it to Coulomb collisions. This new method successfully reduces the growth of particle numbers from the numerical time scale to the physical time scale for Coulomb collisions. We also propose a particle resampling method which reduces the particle number to further improve the efficiency. Various numerical simulations are performed to demonstrate the accuracy and efficiency of the method.

An asymptotic-preserving scheme for linear kinetic equation with fractional diffusion limit

We present a new asymptotic-preserving scheme for the linear Boltzmann equation which, under appropriate scaling, leads to a fractional diffusion limit. Our scheme rests on novel micro-macro decomposition to the distribution function, which splits the original kinetic equation following a reshuffled Hilbert expansion. As opposed to classical diffusion limit, a major difficulty comes from the fat tail in the equilibrium which makes the truncation in velocity space depending on the small parameter. Our idea is, while solving the macro-micro part in a truncated velocity domain (truncation only depends on numerical accuracy), to incorporate an integrated tail over the velocity space that is beyond the truncation, and its major component can be precomputed once with any accuracy. Such an addition is essential to drive the solution to the correct asymptotic limit. Numerical experiments validate its efficiency in both kinetic and fractional diffusive regimes.

Monte Carlo simulation of excitation and ionization collisions with complexity reduction

Abstract Kinetic simulation of plasmas with detailed excitation and ionization collisions presents a significant computational challenge due to the multiscale feature of the collisional rates. In the present work, we propose a complexity reduction method based on atomic level grouping for modeling excitation and ionization collisions. High order of accuracy of the reduction method is realized by allowing an internal distribution within each group. We apply the reduction method to the standard Monte Carlo collision algorithm to model an atomic Hydrogen plasma. Numerical results suggest that the stiffness of the collisional kinetics can be significantly reduced with minimal loss in accuracy.

Coarse-grained Monte Carlo simulation of excitation and ionization collisions

Kinetic simulation of plasmas with detailed excitation and ionization collisions presents a significant computational challenge due to the multiscale feature of the collisional rates. Direct numerical simulation of these collisions using Monte Carlo method requires a large number of samples to resolve all the fast collisional time scales, which is often not necessary and make the simulation very inefficient. In this paper, we present a coarse-graining method to reduce the complexity of detailed excitation and ionization kinetics. The method is then applied to the standard Monte Carlo collision algorithm for simulating a partially ionized Hydrogen plasma. We show that the computational cost can be reduced at minimal loss of accuracy.