Simulation Techniques for HED I: Particle-in-Cell Methods

Transcription

1 2015 HED Summer School Simulation Techniques for HED I: Particle-in-Cell Methods Frank S. Tsung UCLA

2 Special Thanks Viktor Decyk Peicheng Yu < - Slides Michael Meyers Thamine Dalichaouch < - Demos

3 Outline What is PIC? Some examples of large PIC simulations (HED and non-hed) and what are systems which are appropriate for PIC simulations? Various parts of the PIC code, how is each part implemented, show you the limitations of the numerical methods and show you how to design good simulations based on these limitations. Interpolation Particle Orbit Calculation Field Calculation Numerical Cherenkov Radiation - How do we mitigate it so we can study relativistic shocks, and study LWFA s in Lorentz boosted frames

4 Examples of PIC simulations outside of HED PIC models are used in many areas of plasma physics. One area where PIC models have made a large impact is magnetized fusion, and I will show you an example of a large fusion simulation. Gyrokinetic Simulation of Ion Temperature Gradient (ITG) Instability PIC codes are used in space plasmas, nonneutral plasmas, etc.. In this lecture, I will be presenting the PIC models that are used most in the study of HED plasmas and introduce some basic concepts in numerical analysis

5 Examples of PIC simulations in HED Plasmas Electron Plasma Waves in 3D (BEPS simulation relevant to IFE) Laser Solid Interactions in 3D (Ion Accelerations via Fermi Acceleration) Plasma Based Accelerators (Ion Accelerations)

6 References C. K. Birdsall and A. Bruce Langdon, Plasma Physics via Computer Simulation [McGraw-Hill, New York, 1985]. J. M. Dawson, Review of Modern Physics, 55, 403 (1983). R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles [McGraw- Hill, New York, 1981]. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical Recipes, [Cambridge Press, New York, 1986].

7 Particle-in-cell method PIC algorithm Particle du/dt 9.3% time, 216 lines Integration of equations of motion, moving particles F i u i x i Particle dx/dt 5.3% time, 139 lines Field interpolation 42.9% time, 290 lines Interpolation ( E, B ) j F i Δt Current Deposition (x,u) j J j Current deposition 35.3% time, 609 lines Integration of Field Equations on the grid J j ( E, B ) j The particle-in-cell method treats plasma as a collection of computer particles. The interactions does not scale as N 2 due to the fact the particle quantities are deposited on a grids and the interactions are calculated on the grids only. Because (# of particles) >> (# of grids), the timing is dominated by the particle calculations (orbit calculation + current & charge deposition). The grid filters out short-range forces. This approximation is valid because the plasma is fairly collisions and short range forces are not important. In systems where short range interactions are important, a binary collision model can be added back as an additional block after the orbit integration.

8 Inter-particle interactions In PIC models, the E and B fields are functions of the particle quantities such as charge and current densities, and then the particle orbits are updated using the updated electric and magnetic fields. However, the equations for the field can vary depending on the choice of the field model. For today s demos, we will use two different models. Electrostatic (e.g. BEPS): r E = Electromagnetic ~ E = ~ J + r ~ B = r ~ E

9 Note on Notations: Because the systems are discretized in space and time, there are a lot of indices flying around in this talk. So before I begin, I will describe the notations used in this talk. For the rest of the class I will use the following notations: (1) i superscript to indicate time, and j subscript for the j-th particle (for particle quantities such as coordinates or momentum). ~x i j ~x j (t = i t) (2) for grid quantities, the superscript i indicates the time and English subscript indicates the grid coordinate. (3) in some of the codes, the grid quantity is transformed to wave number (k) space via Fast Fourier Transforms, in that case the grid indices are indicated by Greek alphabets, shown below: The smallest wavenumber k, is the mode where the wavelength is exactly the length of the simulation box N x x = L x k x = N x 2 i a,b i (a x, b y) i, i ( k x, k y ) x k y = N y 2 y

10 Short Note on resolution/uncertainty Although there is very little discussion (~2-3 slides) on discrete Fourier transforms, or the a > α notation in the previous slide, the issue of resolution (or uncertainty) is an important one for the rest of this talk. When a system is discretized in space and time like the PIC model, the largest frequency/wavenumber that can be sampled by our choice of dx/dt changes correspondingly. Assuming a system of length L x, discretized into Nx grids, each one of size dx, (and consider a variable φ a, which is defined on these grids). L x = N x dx The lowest wavelength that can be resolved by a system with length L x is: k 0 = 2 the largest frequency, known as the Nyquist frequency, is one which the wavelength is two grids wide. For L x k max = dx = N x 2 k 0 So in k-space, there are also Nx modes, spanning from -kmax to kmax. And the discrete Fourier transform where φ a is transformed to φ α can be described by a matrix multiplication. ~ = M a, ~a a 2 [1,N x ] 2 [ N x /2+1,N x /2] M a, =exp[ik x a ] So, when you introduce discretization in real space, you also limits yourself to the waves behaviors that you can sample. This is something to keep in mind when you design your simulation, but as you will see, we operate far below the Nyquist limit

11 I Interpolating grid quantities on particles/shape functions Another consequence of using grids is that particles cannot be point charges anymore. In particle-in-cell methods, particles are not point charges as point charges cannot be detected by the grid. In order for particles to deposit its charge/ current on the grids (and conversely, for the particles to feel the interactions from the electric & magnetic fields on the grid), the particles need to be of a finite shape. Some common shape functions are: Nearest grid, linear weighting, quadratic spline (these all belong to a family of functions called b-splines, shown on the right), Nearest grid is b-spline of the N-th order is N grid wide (e.g., nearest grid is one grid wide, linear weight is 2 grids wide, etc.) Charge interpolation is written in terms of these shape functions as: (x) = X (q j /V g )S(x x j ) j Having higher order splines can reduce noise, as it allows the code to average over a few grids and this reduces noise from particle statistics Take the system on the right. The effects of particle clumping due to statistical variations can be minimized by using higher order splines. However, this comes at a cost.

12 Particle Shape Function (cont) With the shape function, the grid quantities can be derived from particle positions via the formula: (x) = X j (q j /V g )S(x x j ) By replacing the delta function with the shape function, it can be shown that the cold plasma dispersion becomes (see p. 66 of Birdsall & Langdon).! 2 =! 2 p S(k) 2 k x apple 0.5 (for plasma waves) For the n-th order b-spline, the shape function in Fourier space is: S n (k) = apple sin(k x/2) k x/2 n+1 Therefore, the particle s shape function acts as a filter in Fourier space, and particle s density at large k is effectively less due to the filter effects of the shape function. (A word of caution: because S(x) is a continuous function and the grids are defined at fixed intervals and is subjected to the Nyquist limit, particle shape function at k beyond the Nyquist limit still contributes to the plasma dispersion relation via aliasing, which can be a source of instability (but we do not have time to go into this much detail today but it is a very active topic of research. (we will have a similar requirement for EM waves later)

13 Interpolation Summary Higher order splines are attractive because it reduces statistical fluctuation by spreading the particle over several cells wide it reduces aliasing, and can lead to less numerical problems. However, having larger particle shape introduces finite particle shape S(k) effects into the numerical dispersion relation

14 II Orbit Calculation Leapfrog method Once the current/charge is calculated, and the electric and magnetic fields are calculated, the particle orbits can be advanced via Newton s equation. ( d~xj dt = ~v j d~v j dt = q ~ m E(~x j ) The particle orbits are integrated using the leap-frog method, which is simply the mid-point method for integrating the ODE below. Therefore, to solve the equation above, and to advance the particle s position by one tilmestep, take the velocity at a time between the old time-step and the new time-step, and approximately, the new position becomes: x i+1 j dx dt = v = x i j + v i+1/2 j + Error We will quantify the error in a slide or two.

15 Leap-Frog (cont) However, when the mid-point method is applied to a pair of coupled variables (x,v) which is a function of each other, as shown below: ( d~xj dt = ~v j d~v j dt = q ~ m E(~x j ) then the solution becomes: ~x i+1 j ~v i+3/2 j = ~x i j + ~v i+1/2 j = ~v i+1/2 j + q j m j ( ~ E j+1 (~x j+1 j )) and each particle s position and velocity needs to be staggered by 1/2 step in time in order to perform the integration, and that s how the scheme get its name. The leap-frog method is clean and straight forward with only the electric field, it can incorporate the (vxb) force easily, and the relativistic version of the above equation can also be solved. This is discussed in Birdsall and Langdon and we ll skip it here due to time constrains.

16 Leap Frog Integration (cont) The leap-frog integration method for solving ODE s is remarkable because it is second order accurate in spite of the fact that it only evaluates the time derivative at one point in time. By comparison, both the forward and backward Euler method is only first order accurate. Now, consider trying to march the following ODE in time over a step-size of t. dx dt = f(t) (backward Euler) - 1st order x i+1 = x i + t(f i + O( t)) (forward Euler) 1st order x i+1 = x i + t(f i+1 + O( t)) By evaluating f at the mid-point, the leap-frog scheme is second order accurate. x i+1 = x i + t(f i+1/2 + O( t 2 )) Because this method is second order accurate at each time-step, the cumulative error is first order in t and vanishes as t goes to 0. However, the numerical method is said to be consistent if it is more than second order accurate and is numerically stable. I will talk about the stability condition of the leapfrog scheme next.

17 Stability Condition of the Leap-Frog Method and the Fibonacci Series The stability condition of an ODE solver can be derived by studying the amplifying factors of infinite series. Perhaps some of you have seen an example of this, via the very famous Fibonacci series. x i+2 = x i+1 + x i x 1 =1,x 2 =1 lim i!1 x i = 1+ p 5 2 x i this number, is also known as the Golden Ratio and can be derived by converting the above series into a polynomial (which I will talk about in the next slide). A numerical scheme is said to be stable if the ratio below (called the amplification factor) lies inside the unit circle in the complex plane. This condition insures that the integration errors do not get amplified by the integration scheme. A xi+1 x i apple1

18 Leapfrog stability: I For the leapfrog scheme, let s consider a wave-like equation, i.e., dx dt = i! x Then the leapfrog scheme becomes a series of the form: x i+1 = x i +(i! t)x i+1/2 To calculate the amplification factor, assumes that the series is a function of A (like the Fibonacci series) in the following sense: x i = CA i Then the above series becomes a second order polynomial if you assume (like the Fibonacci series), the series depends on A, as follows: A 2 (i! t)a 1=0

19 Leapfrog Stability: II From Quadratic formula, the amplification A, is: A = (i! t) 2 ± s And therefore the stability condition for the leapfrog method is simply:! t apple 2 1 (i! t) 2 2 Typically, we choose tilmestep of the size: t apple 0.3 [1/! pe ] For magnetized plasmas, the requirement depends on the either the plasma frequency or the cyclotron frequency, depending on which one is higher. t apple 0.3 min(1/! pe, 1/ ce ) For systems with very high intensity lasers (where the wiggle velocity is very large), the step size requirement is even more strict, and is an topic of current research.

20 (Numerical Stability, cont) by contrast, the backward Euler s method is always unstable w.r.t. the model equation, whereas the forward Euler s method is always stable: dx dt = i! x Backward Euler (1st order, unconditionally unstable): x i+1 =(1+i! t)x i A = p 1+! 2 t 2 Forward Euler (1st order, uncoditionally stable): x i = (1 + i! t) xi A = p 1+! 2 t 2

21 III Field Solver for EM field models However, if you want to incorporate laser into your model, you need to solve the full Maxwell s equation. For EM systems, we will will evolve E and B via the two curl equations (we will adopt the OSIRIS normalization of c=1 from this point on), like position and velocity, the electric and magnetic fields of the system can evolve using the leap-frog method using these two equations: Like position and velocity in the particle push, the time derivative of E depends on B and vice versa, so E and B can be advanced in time using the leap-frog method, in the same way as x and v for ~ = r ~ ~ = r ~ E ~ J ~E ~B i i+1 i+2 i+1/2 i+3/2 i+5/2 (time)

22 Courant Condition Independent of the method used to solve Maxwell s equation, all schemes must satisfy the Courant condition. In 1D, it is: i.e., light-wave cannot move more than one cell per timestep. In higher dimensions, it is more complicated: For square (2D) and cubic (3D) grids, the Courant condition for solving Maxwell s equations is: (c) t< 1 p 2 (c) t< x (c) t< x 1 pp i 1/x2 i (c) t< 1 p 3 x respectively

23 FDTD method for EM systems In space, the spatial derivative is done using the midpoint (as it is done in time) because of its good numerical a x a 1 x ( a+1/2 a 1/2)+O( x 2 ) The mid-point method requires the E s and B s to be laid out in a complicated mesh (shown on the right), called the Yee-grid (discovered by Kane Yee in 1966*), but it is fairly straight forward to implement. The x operator is second order accurate, and in Fourier space, instead of (ik), the eigenvalue of the x operator is: x 2sin(k (k) =ik x/2) k x/2 ik (from Wikipedia) apple k2 x 2 (error is second order in x) *

24 Numerical Dispersion Now, putting everything we ve learned so far together, it can be shown that EM waves in this system obeys the dispersion relation: t(!) 2 =! 2 pes(k) 2 + x (k) 2 t (!) = i! 2sin(! t/2)! t/2 which takes into accounts the effects of finite timestep and finite grid size, and also take into account the particle shape. The above equation can be solved numerically and the effects of the grids can be tested before the simulations are performed. (You can also Taylor expand the above equation and solve the polynomial analytically to get some insights). x 2sin(k (k) =ik x/2) k x/2 apple ik k2 x 2

25 Numerical Dispersion (cont) On the top right is the Mathematica prediction of a short pulse propagation in a underdense plasma, n=0.01n c. (dt=0.95 dx). Theory predicts that for kdx<0.3, the agreement between numerical solution and theory (shown in black) is quite good. However, for k x = 0.8, the numerical phase velocity is much smaller than the phase velocity of the real plasma, and the numerical solution deviates from the Exact solution. Lower right shows that OSIRIS results agree very well with theoretical predictions, the pink wiggle on the left of the plot is due to the periodic boundary condition in the Mathematica script. The low k modes of the laser moves to the back of the simulation box but then comes back to the front.

26 Numerical Dispersion (final) So, from the previous slide, one can deduce that the grid acts as a dielectric with different dielectric constants for different k x. (i.e., at large k x, light moves at a velocity slightly slower than it would in a real plasma) This will cause non-physical radiations in systems where there are relativistic beams. We are looking into ways to mitigate this unphysical radiation (numerical Cherenkov)

27 Numerical Cherenkov Radiation Langmuir modes / EM modes coupling

28 Modeling relativistically drifting plasma in EM-PIC code LWFA in Lorentz boosted frame: A way to save time by studying a system in a Lorentz boosted frame. In this case, the plasma is moving near the speed of light. Relativistic collisionless shock: In the simulation below, two relativistically moving plasmas collide with each other, and (nonphysical) radiation develops before the collision happens.

29 hybrid Yee-FFT solver OSIRIS: FDTD solver UPIC-EMMA: spectral solver [k 1 ]= sin(k 1 x 1 /2) x 1 /2 [k 1 ]=k 1 [k 2 ]= sin(k 2 x 2 /2) x 2 /2 [k 2 @x 2 Hybrid solver [k 2 ]= sin(k 2 x 2 /2) x 2 /2 [k 1 ]=k 1 Can be extended to quasi-3d Can use flexible boundary condition in x2 Makes OSIRIS usable.

30 LWFA 3D boosted frame simulation 1.3 GeV 3D simulation NCI elimination applied 3D Simulation parameters grid size: x 1,2,3 =0.196 time step: Boosted frame: t =0.184 x 1 = 15

31 Collisionless relativistic shock Eliminate unphysical energy exchange between EM modes and Langmuir mode log B3 Yee v.s. FFT based solver 2D shock simulation NCI elimination applied 2D Simulation parameters grid size: x 1 =0.5 k0 1 time step: t =0.125 x 1 Drifting velocity: = 20

32 Demos Linear + Nonlinear Landau Damping (BEPS) Beam-Plasma Instability (BEPS) Two Plasmon Instability (OSIRIS)

33 UCLA Particle-in-Cell and Kinetic Simulation Software Center (PICKSC), NSF funded center whose Goal is to provide and document parallel Particle-in-Cell (PIC) and other kinetic codes. Planned activities Provide parallel skeleton codes for various PIC codes on traditional and new parallel hardware and software systems. Provide MPI-based production PIC codes that will run on desktop computers, mid-size clusters, and the largest parallel computers in the world. Provide key components for constructing new parallel production PIC codes for electrostatic, electromagnetic, and other codes. Provide interactive codes for teaching of important and difficult plasma physics concepts Facilitate benchmarking of kinetic codes by the physics community, not only for performance, but also to compare the physics approximations used Documentation of best and worst practices, which are often unpublished and get repeatedly rediscovered. Provide some services for customizing software for specific purposes (based on our existing codes) Key components and codes will be made available through standard open source licenses and as an open-source community resource, contributions from others are welcome.

34 Outline What is PIC? Some examples of large PIC simulations (HED and non-hed) and what are systems which are appropriate for PIC simulations? Various parts of the PIC code, how is each part implemented and how do you design a good simulation based on the numerical scheme used in PIC codes? Interpolation Particle Orbit Calculation Field Calculation Numerical Cherenkov Radiation - How do we mitigate it so we can study relativistic shocks, and study LWFA s in Lorentz boosted frames

35 Extra Slides

36 Boris Pusher: The particle s velocity advance is more complicated when there is a magnetic field (either an external magnetic field or a self-consistent one

37 Boris Pusher (cont)

38 Leap Frog Integration (cont) For the leapfrog scheme, let s consider a wave-like equation, i.e., dx dt = i! x Then the leapfrog scheme becomes a series of the form: x i+1 = x i +(i! t)x i+1/2 To calculate the amplification factor, assumes that the series is a function of A x i = CA i Then the above series becomes a second order polynomial if you assume (like A 2 (i! t)a 1=0 From Quadratic formula, the amplification A, is: A = (i! t) 2 ± s 1 (i! t) 2 2 And therefore the stability condition for the leapfrog method is simply: Typically, we choose tilmestep of the size: For systems with very high intensity lasers (where the wiggle velocity is very t apple 0.3 [1/! pe ]

39 Field Solver for ES field models For studying the plasma wave, the self-consistent magnetic field is ignored and only the electrostatic field is evolved when the particle moves from one step to the next, via the = The above equation can be solved in coordinate space by integrating both sides along x (with ONE boundary condition), or, in most cases, it is solved in k space by the equation. (E x ) = k k = 2 L x In k space, the k=0 mode cannot be solved and is determined by the boundary condition, this is analogous to needing ONE boundary condition in coordinate space. (In higher dimension usually one solves Poisson s equation and then solve for the electric fields)