Introduction to computational plasma physics 雷奕安 62755208，yalei@pku.edu.cn

课程概况 http://www.phy.pku.edu.cn/~fusion/forum/viewtopic.php?t=77 上机 成绩评定为期末大作业

Related disciplines Computation fluid dynamics (CFD) Applied mathematics, PDE, ODE Computational algorithms Programming language, C, Fortran Parallel programming, OpenMP, MPI Plasma physics, space, fusion, … Unix, Linux, …

大规模数值模拟的特殊性 数值计算 数值模拟 大规模数值模拟 数学问题 算法 编程 物理问题 数学模型 算法编程 物理问题及数学模型 相关学科研究人员支持 超级计算机软硬件系统

Contents What is plasma Basic properties of plasma Plasma simulation challenges Simulation principles

What is plasma Partially ionized gas, quasi-neutral Widely existed Fire, lightning, ionosphere, polar aurora Stars, solar wind, interplanetary (stellar, galactic) medium, accretion disc, nebula Lamps, neon signs, ozone generator, fusion energy, electric arc, laser-material interaction Basic properties Density, degree of ionization, temperature, conductivity, quasi-neutrality magnetization

Plasma vs gas Property Gas Plasma Conductivity Very low, insulator Very high, conductor Species Usually one At least two, ion, electron Distribution Usually Maxwellian Usually non-Maxwellian Interaction Binary, short range Collective, long range

Basic properties Temperature Quasi-neutrality Thermal speed Plasma frequency Plasma period

Debye length λD U→0 System size and time Debye shielding

Debye lengths

Plasma parameter Strong coupling Weak coupling

Weakly coupled plasmas

Collision frequency Mean-free-path Collisional plasma (Collisionless) Collisioning frequency

Magnetized plasma Anisotropic Gyroradius Gyrofrequency Magnetization parameter Plasma beta

Simulation challenges Problem size: 1014 ~ 1024 particles Debye sphere size: 102 ~ 106 particles Time steps: 104 ~ 106 Point particle, computational unstable, sigularities

Solution No details, essence of the plasma One or two dimension to reduce the size No high frequency phenomenon, increase time step length Reduce ND, mi / me Smoothing particle charge, clouds Fluidal approaches, single or double Kinetic approaches, df/f

Simple Simulation Electrostatic 1 dimensional simulation, ES1 Self and applied electrostatic field Applied magnetic field Couple with both theory and experiment, and complementing them

Basic model

Basic model

Basic model Field -> force -> motion -> field -> … Field: Maxwell's equations Force: Newton-Lorentz equations Discretized time and space Finite size particle Beware of nonphysical effects

Computational cycle

Equation of motion vi, pi, trajectory Integration method, fastest and least storage Runge-Kutta Leap-frog [ylei@head cp]$ cat b.f90 program abc !implicit double precision (a-h,o-z) x0 = 1 vx0 = 0 y0 = 0 vy0 = 1 !dt = 0.05 read (*,*) dt N = 30/dt do i = 0, N+3 x1 = x0 + vx0*dt y1 = y0 + vy0*dt r = sqrt(x0*x0 + y0*y0) fx = -x0/r**3 fy = -y0/r**3 vx1 = vx0 + fx*dt vy1 = vy0 + fy*dt ! if(mod(i,N/10).eq.2) write(*,*) x0, y0, -1/r+(vx0*vx0+vy0*vy0)/2 x0 = x1; y0 = y1; vx0 = vx1; vy0 = vy1 enddo end [ylei@head cp]$ cat c.f90 N = 50/dt xh0 = (x0+x1)/2; yh0 = (y0+y1)/2 do i = 0, N xh1 = xh0+vx0*dt; yh1 = yh0 + vy0*dt; r = sqrt(xh0*xh0 + yh0 *yh0 ) fx = -xh1/r**3 fy = -yh1/r**3 ! if(mod(i,N/100).eq.0) write(*,*) xh0, yh0, -1/r+(vx0*vx0+vy0*vy0)/2 xh0 = xh1; yh0 = yh1; vx0 = vx1; vy0 = vy1

Planet Problem Forward differencing x0 = 1; vx0 = 0; y0 = 0; vy0 = 1 read (*,*) dt N = 30/dt do i = 0, N+3 x1 = x0 + vx0*dt y1 = y0 + vy0*dt r = sqrt(x0*x0 + y0*y0) fx = -x0/r**3 fy = -y0/r**3 vx1 = vx0 + fx*dt vy1 = vy0 + fy*dt ! if(mod(i,N/10).eq.2) write(*,*) x0, y0, -1/r+(vx0*vx0+vy0*vy0)/2 x0 = x1; y0 = y1; vx0 = vx1; vy0 = vy1 enddo end Forward differencing

Planet Problem ./a.out > data 0.1 $ gnuplot Gnuplot> plot “data” u 1:2

Planet Problem ./a.out > data 0.01 $ gnuplot Gnuplot> plot “data” u 1:2

Planet Problem Leap Frog x0 = 1; vx0 = 0; y0 = 0; vy0 = 1 read (*,*) dt N = 30/dt x1 = x0 + vx0*dt y1 = y0 + vy0*dt xh0 = (x0+x1)/2; yh0 = (y0+y1)/2 do i = 0, N xh1 = xh0+vx0*dt; yh1 = yh0 + vy0*dt; r = sqrt(xh0*xh0 + yh0 *yh0 ) fx = -xh1/r**3 fy = -yh1/r**3 vx1 = vx0 + fx*dt vy1 = vy0 + fy*dt ! if(mod(i,N/100).eq.0) write(*,*) xh0, yh0, -1/r+(vx0*vx0+vy0*vy0)/2 xh0 = xh1; yh0 = yh1; vx0 = vx1; vy0 = vy1 enddo end Leap Frog

Planet Problem ./a.out > data 0.1 $ gnuplot Gnuplot> plot “data” u 1:2

Planet Problem ./a.out > data 0.01 $ gnuplot Gnuplot> plot “data” u 1:2

Field equations Poisson’s equation

Field equations Poisson’s equation is solvable In periodic boundary conditions, fast Fourier transform (FFT) is used, filtering the high frequency part (smoothing), is easy to calculate

Particle and force weighting Particle positions are continuous, but fields and charge density are not, interpolating Zero-order weighting First-order weighting, cloud-in-cell

Higher order weighting Quadratic or cubic splines, rounds of roughness, reduces noise, more computation

Initial values Number of particles and cells Weighting method Initial distribution and perturbation The simplest case: perturbed cold plasma, with fixed ions. Warm plasma, set velocities

Initial values

Diagnostics Graphical snapshots of the history x, v, r, f, E, etc. Not all ti For particle quantities, phase space, velocity space, density in velocity For grid quantities, charge density, potential, electrical field, electrostatic energy distribution in k space

Tests Compare with theory and experiment, with answer known Change nonphysical initial values (NP, NG, Dt, Dx, …) Simple test problems

Server connection Ssh Host: 162.105.23.110, protocol: ssh2 Your username & password Vnc connection In ssh shell: “vncserver”, input vnc passwd, remember xwindow number Tightvnc: 162.105.23.110:xx (the xwindow number) Kill vncserver: “vncserver –kill :xx” (x-win no.)

Xes1 Xes1 document Xgrafix already compiled in /usr/local Xes1 makefile make ./xes1 -i inp/ee.inp LIBDIRS = -L/usr/local/lib -L/usr/lib -L/usr/X11R6/lib64

Clients Ssh putty.exe Vncviewer http://www.phy.pku.edu.cn/~lei/vncviewer.exe Pscp: http://www.phy.pku.edu.cn/~lei/pscp.exe