Outline. 1 Why PIC Simulations Make Sense. 2 The Algorithm. 3 Examples Surface High Harmonics Generation. 4 Extensions Of The PIC Algorithm

Transcription

1 PIC Simulations an Introduction GRK 1203 Meeting February 12-15, 2008, Oelde

2 Outline 1 Simulations Make Sense 2 3 Surface High Harmonics Generation 4 Of PIC

3 Plasma Physics Is Complex Experiment real thing we are actually interested in Plasma diagnostics mostly very indirect Modication of certain parameters may require big eort Analytical ory Yields a profound physical understanding Relies on extensive approximations and assumptions Real experiments often too complex to be fully described by analytical theory Bridging the gap Simulations provide a bridge between experiment and theory

4 How To Simulate? Full kinetic description of a collisionless plasma: Vlasov equation Hydro-Codes p t f + m 1 + xf + F p 2 m pf = 0 Further assume thermal equilibrium Derive momenta equations, easier to solve But: Many interesting plasma phenomena are essentially non-thermal So we stick to the full kinetic description!

5 How To Simulate? Full kinetic description of a collisionless plasma: Vlasov equation Hydro-Codes p t f + m 1 + xf + F p 2 m pf = 0 Further assume thermal equilibrium Derive momenta equations, easier to solve But: Many interesting plasma phenomena are essentially non-thermal So we stick to the full kinetic description!

6 Vlasov PIC It is a challenge to represent f (x, p,t) numerically Direct Vlasov Direct sampling on a 6D Eulerian mesh Straightforward Naturally produces smooth results ly extremely intensive PIC Sampling using Macroparticles: f (x, p) = p W n S(x x p, p p p ) ly eective Intuitive picture of particle clouds

7 Basic Equations Maxwell E = 4πρ B = 0 E = B = 1 c Ḃ c j + 1 c Ė EOM ẋ p = pp m 1+p 2 p ṗ p = F p Fluid elements have to obey equation of motion EOM for macroparticles substitutes Vlasov Macroparticles single electrons/ions How to solve these equations numerically?

8 Basic Equations Maxwell E = 4πρ B = 0 E = B = 1 c Ḃ c j + 1 c Ė EOM ẋ p = pp m 1+p 2 p ṗ p = F p Fluid elements have to obey equation of motion EOM for macroparticles substitutes Vlasov Macroparticles single electrons/ions How to solve these equations numerically?

9 Particles In Cells Fields dened at grid points (cells) E c, B c, ρ c, j c Particles move freely in space x p, p p (taken from Birdsall, Langdon - Plasma Physics via Computer Simulation)

10 Cycle Integrate EOM F p for each particle p Interpolate to particle positions x p, p p for each particle p E c, B c for each cell c Interpolate to cell positions ρ c, j c for each cell c This computational cycle represents one time step Dierent realisations possible Integrate Maxwell Now let us take a walk through the cycle of our code VLPL!

11 Cycle Integrate EOM F p for each particle p Interpolate to particle positions x p, p p for each particle p E c, B c for each cell c Interpolate to cell positions ρ c, j c for each cell c This computational cycle represents one time step Dierent realisations possible Integrate Maxwell Now let us take a walk through the cycle of our code VLPL!

12 Integration Of EOM using the leap-frog scheme (taken from Birdsall, Langdon - Plasma Physics via Computer Simulation) Velocity (respectively momentum p) and position x are dened at staggered timesteps d p = F dt d x = p dt 1+p 2 pnew pold = t Fold xnew xold = p new t 1+p 2 new

13 Accuracy Of Leap-Frog Scheme Test the scheme by applying it to a non-relativistic harmonic potential F = mω 2 0 x Analysis yields: Amplitude is reproduced without mistake Phase error in third order: φ = ω 0 t (ω 0 t) 3 + What about higher order schemes? Big number of particles in simulation Higher order schemes use more storage usually not appropriate for PIC Conclusion simple leap-frog scheme is ideal for PIC

14 Lorentz Force Boris scheme Problem Magnetic force depends on p Leap frog scheme: p not given at the same time as F pnew pold t = q ( E + 1 c pnew + pold 2γ wherein γ is the time-centred relativistic energy B This can indeed be solved for pnew (Boris, 1970) In principle: apply half of the E-eld, then B-eld, then rest of the E-eld )

15 Maxwell Equations I the Yee mesh Use Ampère law to propagate E-eld: Ė = c B 4πj E n+1 E n = ( t) (c ˆ ) B n+1/2 4πj n+1/2, analogue Faraday for B-eld ˆ nite dierence version of curl operator staggered spatial lattice Yee mesh j i 2D geometry B y E x E z E y B z Bx k i B E z x E y B y B x E z 3D geometry j

16 Maxwell Equations II numerical dispersion free scheme k y h y 0 π (a) π k x h x k y h y 0 π (b) k x h x π taken from A Pukhov Straightforward nite dierence translation ( x f ( ) f x+ x/2 f x x/2 / x) of the curl operator ) Stable for (c t) 2 1/ (( x) 2 + ( y) 2 + ( z) 2 Numerical dispersion for short wavelengths (see Fig. (a)) Many problems in laser plasma physics: non-isotropic VLPL also implements a more sophisticated scheme Stable for c t x No articial dispersion in x-direction (see Fig. (b))

17 Maxwell Equations III divergence equations Substitute divergence equations ( E = 4πρ, B = 0 ) by the charge continuity equation ρ + j = 0 (1) = If divergence equations satised initially, they will automatically remain In VLPL, (1) is guaranteed by the proper (ρ, j) interpolation scheme

18 Interpolating ρ c Imagine macroparticles as massive cubes with uniform charge distribution edge length = grid step Distribute charge to the cells appropriately linear interpolation scheme Figure shows example in 1D taken from Birdsall / Langdon (a) shows the macroparticle inside the grid (b) shows the charge contained in grid cell x i as macroparticle passes by

19 Interpolating j c And Field Quantities Important for rigorous charge and energy conservation! Dene j c half timesteps shifted in relation to ρ c and at the cell boundaries Now follow the particle trajectories t j c = da dt vw p S p = da c 0 to perfectly fulll the continuity equation c xnew xold d xw p S p For optimal energy conservation, E-eld interpolation must be consistent with the current weighting

20 Input Parameters for a PIC run Initial conditions Particles Fields plasma or electron beams can be initialised inside the simulation box laser pulses may be initialised inside the box various shapes possible and readily implemented Boundary conditions Can be reecting, periodic, or absorbing both for particles and elds Particles or laser pulses may also enter the simulation box

21 Parallelisation For simple yet eective parallelisation, the simulation box is split in partitions A special boundary condition allows for data exchange between the partitions In the VLPL code, parallelisation is realised using MPI (message passing interface)

22 Diagnostics Because PIC produces huge amounts of data, it is usually not possible to store everything Before the simulation one must think of what to save: Time histories E.g. record the E- and B-elds at certain points in space Later one may Fourier-transform to obtain spectra Snapshots Movies Save all data at a certain time step Take a cross section and save periodically

23 Surface High Harmonics Generation Relativistic laser reected at an overdense plasma surface Reected light spectrum contains multiples of the laser frequency (harmonics) Power law spectrum found by Baeva, Gordienko, Pukhov Results conrmed rst by PIC simulation, then by experiment PIC simulation provides vivid insight to what's happening

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44 Pictures and Simulation by Anupam Karmakar Electron beam propagating through a plasma provokes instability PIC brings not only marvellous pictures, but also deep insight into the physics E.g., collisions and temperature can be switched on or o at will to study their eects separately

45 Picture by A Pukhov Ultra-short, ultra-relativistic laser pulse propagating through tenuous plasma In the wakeeld a bubble emerges, wherein electrons are accelerated to very high energies Predicted by PIC, conrmed by experiments

46 Of PIC small modications Ionisation Using a Monte Carlo method, implemented for VLPL by A Karmakar Allows for simulation of laser interaction with high Z ions Collisions Using a Monte Carlo method, implemented for VLPL by A Karmakar Interesting e.g. for instability Lorentz transformation of initial conditions and results Changing the frame of reference allows to simulate some cases more eectively

47 Of PIC big modications Hybrid codes A part of the plasma is represented hydrodynamically, the non-thermal part by PIC Allows for simulation of non-thermal high density plasmas without resolving the plasma frequency Implemented in 1D by A Karmakar (One dimensional electromagnetic relativistic PIC-hydrodynamic hybrid simulation code H-VLPL (Hybrid Virtual Laser Plasma Lab), in press, Computer Physics Communication), 3D version planned Other interesting ideas: Quasi-static PIC codes Adaptive codes...

48 PIC simulation is an indispensable tool for modern plasma physics. algorithm is a concatenation of basically simple, traceable steps. Clever extensions allow to simulate even cases that are not covered by classical PIC.

49 For Further Reading Appendix For Further Reading C K Birdsall and A B Langdon. Plasma Physics via Computer Simulation. IOP Publishing, A Pukhov. Three-Dimensional Particle-in-Cell Simulations of Relativistic Laser-Plasma Interactions. Lecture Notes. Osaka University, 1999.