High-Order Semi-Lagrangian WENO Schemes Based on Non-polynomial Space for the Vlasov Equation

In this paper,we present a semi-Lagrangian(SL)method based on a non-polynomial function space for solving the Vlasov equation.We fnd that a non-polynomial function based scheme is suitable to the specifcs of the target problems.To address issues that arise in phase space models of plasma problems,we develop a weighted essentially non-oscillatory(WENO)scheme using trigonometric polynomials.In particular,the non-polynomial WENO method is able to achieve improved accuracy near sharp gradients or discontinuities.Moreover,to obtain a high-order of accuracy in not only space but also time,it is proposed to apply a high-order splitting scheme in time.We aim to introduce the entire SL algorithm with high-order splitting in time and high-order WENO reconstruction in space to solve the Vlasov-Poisson system.Some numerical experiments are presented to demonstrate robustness of the proposed method in having a high-order of convergence and in capturing non-smooth solutions.A key observation is that the method can capture phase structure that require twice the resolution with a polynomial based method.In 6D,this would represent a signifcant savings.

Global Weak Solutions to a Fluid-particle System of an Incompressible Non-Newtonian Fluid and the Vlasov Equation

The purpose of this work is to investigate the existence and uniqueness of weak solutions to the initial-boundary value problem for a coupled system of an incompressible non-Newtonian fluid and the Vlasov equation.The coupling arises from the acceleration in the Vlasov equation and the drag force in the incompressible viscous non-Newtonian fluid with the stress tensor of a power-law structure for p≥11/5.The main idea of the existence analysis is to reformulate the coupled system by means of a so-called truncation function.The advantage of the new formulation is to control the external force term G=-∫Rd(u-v)fdc(d=2,3).The global existence of weak solutions to the reformulated system is shown by using the Faedo-Galerkin method and weak compactness techniques.We further prove the uniqueness of weak solutions to the considered system.

Conservative Semi-Lagrangian Finite Difference WENO Formulations with Applications to the Vlasov Equation

In this paper,we propose a new conservative semi-Lagrangian(SL)finite difference(FD)WENO scheme for linear advection equations,which can serve as a base scheme for the Vlasov equation by Strang splitting[4].The reconstruction procedure in the proposed SL FD scheme is the same as the one used in the SL finite volume(FV)WENO scheme[3].However,instead of inputting cell averages and approximate the integral form of the equation in a FV scheme,we input point values and approximate the differential form of equation in a FD spirit,yet retaining very high order(fifth order in our experiment)spatial accuracy.The advantage of using point values,rather than cell averages,is to avoid the second order spatial error,due to the shearing in velocity(v)and electrical field(E)over a cell when performing the Strang splitting to the Vlasov equation.As a result,the proposed scheme has very high spatial accuracy,compared with second order spatial accuracy for Strang split SL FV scheme for solving the Vlasov-Poisson(VP)system.We perform numerical experiments on linear advection,rigid body rotation problem;and on the Landau damping and two-stream instabilities by solving the VP system.For comparison,we also apply(1)the conservative SL FD WENO scheme,proposed in[22]for incompressible advection problem,(2)the conservative SL FD WENO scheme proposed in[21]and(3)the non-conservative version of the SL FD WENO scheme in[3]to the same test problems.The performances of different schemes are compared by the error table,solution resolution of sharp interface,and by tracking the conservation of physical norms,energies and entropies,which should be physically preserved.

A fast convergence fourth-order Vlasov model for Hall thruster ionization oscillation analyses

A 1D1 V hybrid Vlasov-fluid model was developed for this study to elucidate discharge current oscillations of Hall thrusters(HTs).The Vlasov equation for ions velocity distribution function with ionization source term is solved using a constrained interpolation profile conservative semiLagrangian method.The fourth-order weighted essentially non-oscillatory(4 th WENO)limiter is applied to the first derivative value to minimize numerical oscillation in the discharge oscillation analyses.The fourth-order accuracy is verified through a 1 D scalar test case.Nonoscillatory and high-resolution features of the Vlasov model are confirmed by simulating the test cases of the Vlasov–Poisson system and by comparing the results with a particle-in-cell(PIC)method.A1 D1 V HT simulation is performed through the hybrid Vlasov model.The ionization oscillation is analyzed.The oscillation amplitude and plasma density are compared with those obtained from a hybrid PIC method.The comparison indicates that the hybrid Vlasov-fluid model yields noiseless results and that the steady-state waveform is calculable in a short time period.

Landau damping in a bounded magnetized plasma column

The damping decrement of Landau damping and the effect of thermal velocity on the frequency spectrum of a propagating wave in a bounded plasma column are investigated.The magnetized plasma column partially filling a cylindrical metallic tube is considered to be collisionless and non-degenerate.The Landau damping is due to the thermal motion of charge carriers and appears whenever the phase velocity of the plasma waves exceeds the thermal velocity of carriers.The analysis is based on a self-consistent kinetic theory and the solutions of the wave equation in a cylindrical plasma waveguide are presented using Vlasov and Maxwell equations.The hybrid mode dispersion equation for the cylindrical plasma waveguide is obtained through the application of appropriate boundary conditions to the plasma-vacuum interface.The dependence of Landau damping on plasma parameters and the effects of the metallic tube boundary on the dispersion characteristics of plasma and waveguide modes are investigated in detail through numerical calculations.

Enhanced electron–positron pair production by frequency chirping in one-and two-color laser pulse fields

Enhanced electron–positron pair production by frequency chirping in one- and two-color laser pulse fields is investigated by solving the quantum Vlasov equation. A small frequency chirp shifts the momentum spectrum along the momentum axis. The positive and negative frequency chirp parameters play the same role in increasing the pair number density. The sign change of the frequency chirp parameter at the moment t = 0 leads the pulse shape and momentum spectrum to be symmetric, and the number density to be increased. The number density of produced pairs in the two-color pulse field is much higher than that in the one-color pulse field and the larger frequency chirp pulse field dominates more strongly. In the two-color pulse fields, the relation between the frequency ratio of two colors and the number density is not sensitive to the parameters of small frequency chirp added in either a low frequency strong field or a high frequency weak field but sensitive to the parameters of large frequency chirp added in a high frequency weak field.

Enhanced electron positron pair creation by the frequency chirped laser pulse

Based on the quantum Vlasov equation, the effect of frequency chirp on electron-positron pair production is investigated. The cycle parameter, which characterizes the laser field cycle degree within the pulse, is also considered. In both supercycle and subcycle laser pulses the frequency chirp can greatly enhance the momentum distribution function of created pairs and the pair number density. The pair number density created by a supercycle laser pulse is larger than that by a subcycle pulse under the same laser frequency and chirping. There exists an optimal cycle parameter corresponding to the maximum value of the created pair number density for different chirp rates. It is found that the pair number density is sensitive/insensitive to chirping rate when the cycle parameter lies below/above the optimal one.

ON THE EXISTENCE OF ALMOST GLOBAL WEAK SOLUTION TO MULTIDIMENSIONAL VLASOV-POISSON EQUATION

The authors prove the existence of almost global weak solution to multidimensional Vlasov Poisson equation with a class of Randon measure as initial data.

An Accurate Numerical Scheme for Maxwell Equation with CIP-Method of Characteristics

A new multi-dimensional scheme for the Maxwell equations is established by the CIP method in combination with the method of characteristics(CIP-MOC).In addition,the CIP-MOC can be extended to arbitrary grid system by the Soroban grid without losing the third-order accuracy.With the accuracy fixed,the grid points required for the CIP are 40 times less than the conventional schemes like the FDTD in three dimensions.Numerical solutions obtained by the CIP-MOC are compared with analytical solution and the FDTD in plane-wave scattering by a perfectly-conducting circular cylinder,and the CIP-MOC agrees very well with analytical solutions.The Soroban grid is also applied to the Vlasov equation that describes the kinematics of plasmas that is frequently combined with the Maxwell equation.The adaptively moving points in velocity space are similar to the particle codes but can provide accurate solutions.

Direct Vlasov Solvers with High-Order Spectral Element Method

This paper presents the development of parallel direct Vlasov solvers using the Spectral Element Method(SEM).Instead of the standard Particle-In-Cell(PIC)approach for kinetic space plasma simulation,i.e.solving the Vlasov-Maxwell equations,the direct method has been used in this paper.There are several benefits to solve the Vlasov equation directly,such as avoiding noise associated with the finite number of particles and the capability to capture the fine structure in the plasma,etc.The most challenging part of direct Vlasov solver comes from high dimension,as the computational cost increases as N2d,where d is the dimension of the physical space.Recently due to fast development of supercomputers,the possibility of high dimensions becomes more realistic.A significant effort has been devoted to solve the Vlasov equation in low dimensions so far,now more interests focus on higher dimensions.Different numerical methods have been tried so far,such as finite difference method,Fourier spectral method,finite volume method,etc.In this paper SEM has been successfully applied to construct these solvers.SEM has shown several advantages,such as easy interpolation due to local element structure and long time integration due to its high order accuracy.Domain decomposition in high dimensions have been used for parallelization,these include scalable parallel 1D and 2D Poisson solvers.Benchmark results have been shown and simulation results have been reported for two different cases:one dimension(1P1V),and two dimensions(2P2V)in both physical and velocity spaces.

GLOBAL WEAK SOLUTIONS OF THE RELATIVISTIC VLASOV-KLEIN-GORDON SYSTEM IN TWO DIMENSIONS

In this paper we consider the system of classical particles coupled with a Klein-Gordon field in two dimensions. We establish a-priori-bounds on the solutions of this system with initial data satisfying a size restriction derived from conservation of energy. This result, together with the smoothing of ＂velocity averaging＂, yields the existence of global weak solutions to the corresponding restricted initial value problem. The size restriction is necessary since energy of the system is indefinite. Finally, we show that the weak solutions preserve the total mass.

Theoretical investigation of a backward wave oscillator

From the linear Vlasov equation, the theoretical investigation on relativistic backward wave oscillator is performed. The relationship between the microwave power and the guiding magnetic field, which accords with the results of the particle simulation and experiments, is deduced.

Dynamically assisted pair production for scalar QED by two fields

By solving the quantum Vlasov equation, the dynamically assisted pair production for scalar quantum electrodynamics （QED） is investigated. It is verified that this mechanism still holds true for boson pair production. Two combinations of two electric fields having different time scales under various time delays are considered; it is found that the oscillations of the momentum spectrum and the number density of created bosons decrease with increasing time delay, and the latter has a maximum value when the time delay equals zero. Furthermore, the differences in vacuum pair production between bosons and fermions are also studied, and they are helpful for distinguishing the created bosons from fermions.

Electron-positron pair production in a strong asymmetric laser electric field

By solving the quantum Vlasov equation, electron positron pair production in a strong electric field with asymmetric laser pulses has been investigated. We consider three different situations of subcycle, cycle and supercycle laser pulses. It is found that in asymmetric laser pulse field, i.e.. when the pulse length of one rising or falling side is fixed while the pulse length of the other side is changed, the pair production rate and mnnber density can be significantly modified comparable to symmetric situation. For each ca,se of these three different cycle pulses, when one side pulse length is constant and the other side pulse length becomes shorter, i.e., the whole pulse is compressed, the more pairs can be produced than that in tile vice versa case, i.e., the whole pulse is elongated. In compressed pulse case there exists an optimum pulse length ratio of asylnmetric pulse lengths which makes the pair number density maximunn. Moreover, the created maximum pair number density by subcycle pulse is larger than that by cycle or/and supercycle pulse. In elongated pulse case, however, only for supercycle laser pulse the created pairs is enhanced and there exists also an optimum asymmetric pulse length ratio that maximizes the pair number density. On the other hand. surprisingly, in both cases of subcycle and cycle elongated laser pulses, the pair number density is monotonically decreasing as the asymmetry of pulse increases.

Electron-positron pair production in the low-density approximation

Electron-positron pair creation is studied in the low-density approximation by solving the quantum Vlasov equation exactly and the mapping equation approximately. The simpler mapping equation is an approximate treatment of the quantum Vlasov equation in which the continuous external field is regarded as a series of delta kicks. Our study indicates that this new treatment is appropriate because the results of the two methods are in good agreement with each other. However, as the period number increases, interference and a complicated structure in the momentum distribution are observed. Furthermore, we also obtain the square power law relation of the number density to the applied electric field strength.

A Coulomb explosion theoretical model of femtosecond laser ablation materials

In this paper,a kinetic theory of Vlasov equation is proposed to depict electron and ion's nonequilibrium transport processes in a femtosecond time scale.A Coulomb explosion model of femtosecond laser ablation of materials is proposed and numerically simulated.The mechanism of surface Coulomb explosion induced by self-consisted electric field and the impact of laser parameters on the ablation of materials are quantitatively analyzed.The ablation depths calculated by the model are in good agreement with the experimental results.It is shown that,the intensity of self-consisted electric field generated on the dielectric material's surface is much greater than that generated on the metal or the semiconductor material's surface,and Coulomb explosion ablation is more easily to occur on the dielectric material's surface.

Electron-Positron Pair Production in a Strong Laser Field Enhanced by an Assisted High Frequency Weak Field

Electron-positron pair production in a strong laser field enhanced by an assisted high frequency weak field is investigated by solving the quantum Vlasov equation.The average and residual pair number densities are obtained for sinusoid electric field and it is found that the high frequency assisted weak field will enhance pair production significantly.There exists an optimal frequency of assisted field that makes the pair production number density get a maximum one,which is a few orders of higher than that without assisted field.We also discuss the other possible assisted fields.