Kinetic Simulation Method For Typical Problems In Plasma Physics

Plasma is the fourth state of matter after solid,liquid and gas.It exists widely in the universe from auroras to nebulae,from neon lights to the Sun so the study of its behaviors is important.Particularly in the controlled fusion study,we are to confine the plasma with high temperature for enough long time.It requires an in-depth understanding of motion of plasma in complex configuration with electromagnetic field.Compared with single-particle orbit method and magnetic hydrodynamic method,kinetic method can give a more fined description of plasma bahaviours.The key problem of kinetic theory of plasma physics is to solve and analyze the kinetic equation.The kinetic equation without Coulomb collisions is called Vlasov equation.The relaxation time collision term(BGK collision term),Landau collision term and Fokker-Planck collision term are commonly used to depict the Coulomb collisions.The electromagnetic field generated by plasma motion is depicted by Maxwell equations.There are seven free degrees(x,v,t)in kinetic equation.Because of highdimensional and multi-physical properties,it is difficult to solve it analytically.So numerical simulation plays an important role in the study of plasma physics.Particle-In-Cell method and semi-Lagrangian method are most popular methods in kinetic study of plasma physics.Unified gas-kinetic scheme is proposed to solve neutral gas problem originally and it is then expanded to plasma simulation.In this paper,we propose a series of numerical methods based on gas-kinetic scheme to solve typical problems in plasma physics such as stimulated Raman scattering and Z-pinch.Stimulated Raman scattering is a typical instability with three waves in laserplasma interaction study.We are to study the energy evolution of three waves including pump wave,scattering wave and electrostatic wave and the generation and absorption of relativistic electrons.To study collisional stimulated Raman scattering,we use relativistic Vlasov-BGK equation coupled with Maxwell equations to depict it and propose a modified gas-kinetic scheme to solve them.High order spline reconstruction and interpolation is used to reduce the error of energy of three waves in the long-time evolution.To study the two-dimensional stimulated Raman scattering and its cascade process,we generalize modified gas-kinetic scheme into 2D case.With the application of Strang-splitting strategy,Vlasov-BGK equation is divided into transport-collision part and acceleration part.Time evolution solution is used in the design of numerical flux of transport-collision part and quadratic B-spline reconstruction is adopted.The acceleration part is solved by semi-Lagrangian method with cubic B-spline interpolation.In the laser-plasma interaction study,high order reconstruction is needed to describe its strong non-linearity caused by Lorentz factor and multi-physics property,but non-physical negative values of distribution function will be introduced at the same time.So we propose a positive preserving gas-kinetic scheme to solve relativistic Vlasov-BGK equation with strict proof,and apply it on the study of stimulated Raman scattering and relativistic modulational instability.In magnetic confinement fusion study,many configurations are axisymmetric,such as Tokamak,Z-pinch.It would be convenient to study them in cylindrical coordinates.So we propose a cylindrical gas-kinetic scheme to solve Vlasov-BGK equation in cylindrical coordinates and prove its asymptotic preserving property.It is applied on the simulation of charge separation of plasma edge model and Z-pinch configuration.Numerical results demonstrate our scheme can capture the process from non-equilibrium state to equilibrium state by Coulomb collisions.