Solving Vlasov Equation for Beam Dynamics Simulation

This paper presents the development of parallel direct Vlasov solvers using the Spectral Element Method (SEM). There are several benefits to the direct method over the standard PIC approach for solving the Vlasov equation, such as avoiding the noise associated with a finite number of particles and the capability to capture the fine structure in the plasma. The most challenging aspect of the direct Vlasov solver is the size of the problem where the computational cost increases as d N 2 , where d is the dimension of the physical space and N the number of mesh nodes per dimension. We show that the SEM method has 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 is used for parallelization, includes scalable parallel 2D Poisson solvers. Benchmarks and simulation results are reported in two dimensions in both the physical and velocity spaces (2P2V). INTRODUCTION Plasma and charged particle simulations have great importance in science. There are three different approaches to simulate plasmas: the microscopic model, the kinetic model and the fluid model. In the microscopic model, each charged particle is described by 6 variables (x, y, z, z y x v v v , , ). Therefore, for N particles, there are 6N variables in total. This requires solving the Vlasov equation in 6N dimensions, which exceeds the capability of current supercomputers for very large N. On the other end is the fluid model which is the simplest because it treats the plasma as a conducting fluid with electromagnetic forces exerted on it. This leads to solving the Magneto-hydrodynamics (MHD) equations in 3D (x, y and z). MHD solves for the average quantities, such as density and charge, which makes it difficult to describe the fine structure in the plasma. Due to computer speed limitations, MHD is currently the most popular approach in plasma simulations. Between these two models is the kinetic model, which solves for the charge density function by solving the Boltzmann or Vlasov equations in 6 dimensions (x, y, z, z y x v v v , , ). The Vlasov equation describes the evolution of a system of particles under the effects of self-consistent electromagnetic fields. This paper deals with the kinetic model. There are two different ways to solve the kinetic model. The most popular one is to represent the beam bunch by macro particles and push the macro particles along the characteristics of the Vlasov equation. This is the so called Particle-In-Cell (PIC) method, which utilizes the motion of the particles along the characteristics of the Vlasov-equation using a Lagrange-Euler approach [1, 2]. In principle, it simplifies the Partial Differential Equation (PDE) to an Ordinary Differential Equation (ODE). The interaction between charged particles, which is called the space charge force, is handled by solving Poisson’s equation. Then the electric field from the potential solution can be computed. The PIC method has the advantages of speed and easy implementation, but similar to MHD, it is hard to calculate fine structures in the plasma. Furthermore, there is noise associated with the finite number of particles in the simulation. This noise decreases very slowly, as N / 1 , when the number of particles N is increased. The other way to solve the kinetic model is to solve the Vlasov equation directly. This can overcome the shortcomings of the PIC method, but due to the high dimensional nature of the Vlasov equation, numerical simulations have generally been conducted in low dimensions such as 1P1V or the axisymmetric case [3, 4]. Recently, 2P2V simulations have been reported [5]. We have applied SEM which can achieve high order accuracy than [5] and developed scalable Poisson and Vlasov solvers to make use of the BG/P supercomputer at ANL. VLASOV EQUATION The distribution function ) , , ( t v x f r r in phase space is governed by the Vlasov equation. In beam dynamics, a simplified model can be deduced in 2P2V form as a paraxial model based on the following assumptions: • The beam is in a steady-state: All partial derivatives with respect to time vanish; • The beam is sufficiently long so that the longitudinal self-consistent forces can be neglected; • The beam is propagating at a constant velocity b v along the propagation axis z; • Electromagnetic self-forces are included; • , and ~ ), , , ( b y x b z z y x p p p p p p p p p << = r where b b mv p γ = is the beam momentum. It follows in particular that 2 / 1 2 2 ) 1 ( , ) / ( − − = ≈ = ≈ b b b b c v β γ γ β β • The beam is narrow: the transverse dimensions of the beam are small compared to the characteristic longitudinal dimension. The paraxial model can be written as: 0 ) ) , ( 1 ( 2 = ∇ ⋅ × + + Φ ∇ − + ∇ ⋅ + ∂ ∂ f B v v E mv q f v v z f