PIC solvers for Intense Beams in Synchrotrons

Transcription

1 PIC solvers for Intense Beams in Synchrotrons Oliver Boine-Franenheim TU Darmstadt and GSI

2 Contents o Application and challenges for beam simulations with space charge o D electrostatic Particle-In-Cell (PIC) simulation scheme for beams o Review: umerical noise in PIC codes and related emittance growth o Digital filters o Symplectic PIC and application to D beams o Summary and Outloo

3 Space Charge Simulations in Synchrotrons (in the rest system of the beam) E r B φ Space charge tune shift: Tune footprint (CER PS simulation, Franchetti 003) Bunch (rms length: 0.1 m dc) Bunch length >> pipe diameter The transverse space charge force is the main intensity limiting effect in in the FAIR synchrotrons at GSI and in other high current synchrotrons. Time scales: turns (1 ms - 1 s) -> Emittance growth, Beam loss (< 1%) Challenge: Control of numerical emittance growth! Space charge collaboration (CER, GSI, FAL, SS, KEK, ) o Codes for long-term simulations -> Effect of stochastic noise! o (PIC) codes used for FAIR at GSI: pyorbit, icromap, PATRIC 3

4 PIC simulation scheme (for D beams) q : beam particle charge Q= q : macro particle charge : number of beam particles : number of macro-particles qe (, y, s) κ() s = 0 i i i i γ 0mv0 qey (, y, s) yi + κ() s yi = 0 γ i i 0mv0.5D electric fields: bunches are long (m) Compared to their transverse width (cm). Error sources: 1) artificial collisions of macro-particles Q and test particles q. (Coulomb s law) Δs -> energy conserving. ) Grid/stochastic heating Depending on particle shape S the macro-particles have a finite width: Δ Δ 4

5 What physics system does a PIC code represent? The system we would lie to solve: Collision-free Vlasov-Poisson (V-P) system f f f + +F = 0 t (Vlasov equation) ò0 φ = ρ(, y) (Poisson equation) ρ (, y) = q f (, ) d F =, () s q κ y φ(, y,) s γ mv S = f ln fd d = const. 0 0 (charge density) (AG focusing) (space charge) (Entropy) V-P particles : phase space marers What in fact we solve is (forget about the grid for a moment): F F F + +F = 0 t (Klimontovich equation) F(, ) = S( ) δ ( ) = 0 -> Artificial collisions and related noise. -> oise level is much stronger than the Schotty noise in real beams. S4 = FlnFd d = const. (Entropy) [1] Winum et al., Verification and convergence properties of PIC codes (014) 5

6 Artificial collisions in a D computer beam In a D beam the beam macro-particles are rods: Collision angle independent on b! All particles with relative velocities less than Debye length: are deflected by angles > beam rod: Q = Q L collision angle: Friction force on a test particle: F v = m v = m d vdbu f v p ( ) ν ( )sin ( θ /) ν Δ 1 λd λd ν ln Δ (D collision rate for finite -sized macro particles) (3D collision rate) impact parameter b: Test beam ion: q F qq = πε r 0 (D Coulomb force) Relation between friction and diffusion in a system with energy conservation: T B D = ν (Einstein relation) m T(s) = T + T / B ( y) T mc ε ˆ β (grid spacing) [1] Boine-Franenheim, Hofmann, Strucmeier, Appel, IA (015) 6

7 Entropy and emittance growth due to artificial collisions in computer beams J. Strucmeier, Stochastic effects in real and simulated charged particle beams, Phys. Rev. ST-AB 000 Entropy/Emittance growth for anisotroptic beams: T T T 1 dε 1 ( T Ty) B D = ν = Bν m ε dt T T (4D emittance) Emittance (growth) PATRIC (D PIC): Emittance growth after FODO cells ε = ε 0, 0, y ν y o μ0 = 60 sc Δμ = 10 o Emittance (growth) y -> Emittance growth due to AG focusing in a FODO channel! -> o growth for constant focusing (CF) and for FODOXX channels. FODOXX: κ () () ˆ ˆ s = κy s β = βy CF: κ = κ = const. ν y 1 / macro-particle number [1] Hofmann, Boine-Franenheim, PRAB (015) 1 / macro-particle number Emittance growth in CF/FODOXX channels. -> Obviously there is additional heating! 7

8 Short comment on the effect of the grid spacing on emittance growth PATRIC (D PIC): Emittance growth after FODO cells Chosen grid size in eamples ( =64) PATRIC (D PIC): FODO vs. CF Chosen grid size in eamples ( =64) Artificial collisions Artificial collisions (grid points) (grid points) 8

9 Stochastic or grid heating ( inelastic collisions ) y Random wal in grid 1 q induced error fields: T n ( ) ( s) m δ E Δ umerical heating due to grid! q δ = δeδs m T B D >ν m δ E Random grid induced electrical error field: rounding, time step, finite differences, A non-conservative force (-> heating) is introduced by grid undersampling. -> Standard PIC integrators are non-symplectic ν ν T ν Stochastic heating in computer plasmas: [1] Hocney, easurement of collision and heating times in a D computer plasma, J. Comput. Phys. (1971) Application to computer beams: [] Kesting, Franchetti, Propagation of numerical noise in PIC, PRAB (015) time steps oise is not bad, the grid (undersampling) is 9

10 Digital filters for PIC + Reduce noise at short wavelengths (smoothing of the charge density). + Relatively fast. - Can lead to non-physical results and numerical instabilities - (on-) Energy conservation (e.g. damping of modes) ρ i, = ρ i, + S( side terms) + K( corner terms) + 4( S + K) (, S, K ) = iddle, side and corner weights (grid points) Eamples: Hollow 8-point filter (0,1,1): Studied in []. Has negative spectral representation. Binomial 9-point filter (4,,1): Proposed for PIC codes in [1]. Digital filters: [1] Birdsall, Langdon, Plasma Physics via Computer Simulation (004) [] Hocney, Eastwood, Computer Simulation using Particles (1988) [3] Verboncoeur, PIC for plasmas: Review and Advances, Plasma Phys. Control. Fusion (005) [4] Vay et al., umerical methods for instability mitgation in, J. Comput. Phys. (011) Wavelet denoising: [4] Gassama, Sonnendrücer, Wavelet denoising for postprocessing of D PIC codes, ESAI (007) [5] Terzic et al., PIC simulations with a wavelet based Poisson solver, PRAB (007) 10

11 Eample results for FODO channel with filters PATRIC D: umerical emittance growth in a FODO channel after cells with 9-point filter o sc μ0 = 60, Δμ = 10 o After 500 cells: no filter vs. with filter With digital filters: -> Emittance growth rate is slower (appro. factor 10). -> However, emittance growth is not the only benchmar -> resonances, 11

12 Symplectic Particle-In-Cell Conventional PIC: Total momentum is conserved, but not energy (grid heating). Energy conserving PIC [3]: omentum is not conserved (self-force on the grid). -> PIC codes are usually not symplectic An ideal beam particle integrator with space charge would conserve: Phase space (Symplectic), Emittance, Entropy, Energy (only for constant focusing) ulti-symplectic integrators offer bounded variations of those constants. Starting point is usually the Lagrangian (here for a D beam with space charge): 1 1 q 0 L d0dy0d0dy0f( 0, y0, 0, y0) (( ) ( y ) ) ( ) ( ) κ y ddy E φ ò = + 0γ0β0 E φ φ 0γ0β 0 Equations of motion (for particles and fields) from Euler-Lagrange equations: L d L = 0 ξ ds ξ q + κ () s = E γ β [1] Webb, A spectral canonical electrostatic algorithm, Plasma Phys. Control. Fusion (016) [] Shadwic et al., Variational formulation of macro-particle plasma simulation algo., PoP (014) [3] Langdon, Energy-Conserving plasma simulation algorithms, J. Comput. Phys. (1973) [4] Qin, et al., Canonical symplectic PIC method for long term simulations.., ariv (015) φ y κ() s = E0γ0β0 q φ y ò0 Δ φ = ρ 1

13 Spectral (grid-less) computer beam Lagrangian Beam macro-particle distribution: Spectral space charge potential f(, ) = w S( ) δ ( ) φ( ) = φ ep( i ) = 0 Resulting discrete Lagrangian: L = W W W W D AG sc field W 1 ( ) q 1 W = w S( ) φ ep( i ) Wfield = φφ γ β = 0 E γ β = w sc = 0 E (inetic) (space charge potential) (space charge field energy) Equations for macro-particles and spectra field) from Euler-Lagrange equations: LD ξ d LD = 0 ds ξ iq i + κ() s = φ S( ) e E γ β ε φ = ρ 0 Spectral (grid-less) PIC for plasmas: [1] Webb, A spectral canonical electrostatic algorithm, Plasma Phys. Control. Fusion (016) [] Decy, Spectral PIC codes, ISSS (011) [3] Huang et al., Grid instability and spectral fidelity of ES PIC codes, ariv: (016) 13

14 Spectral algorithm for D beams Spectral charge density: Space charge potential: Space charge ic: n+ 1 i ρ = ws( ) e ε φ = ρ 0 iq Δ = Δ * i n s ( ) φ S e E0γ0β0 Time step ( ): Δs ( sn, sn+ 1) + Δ = y y y y + y Δ n n Tent particles shape: Δ 1 / Δ, Δ S( ) = 0, >Δ Δ 1 S ( ) = ds ( )ep( i) π D spectral particle shape: S( ) = sinc ( Δ / ) sinc ( Δy / ) Algorithm derived from a grid-less macro-particle Lagrangian. We epect the system to follow: y T B D ν m 14

15 Tests: Spectral PIC vs. Conventional PIC Emittance (growth) PATRIC D: umerical emittance growth in a FODO channel after 500 cells. o sc μ0 = 60, Δμ = 10 o Spectral/grid points: = = 64 Emittance (growth) Spectral PIC only. Bounded emittance 1 / macro-particle number 1 / macro-particle number oise still present (in -space), but no emittance growth. However, spectral PIC is (still) much slower: T α spectral α cells 0 50 α grid Remar: For a python/numpy/cython implementation and on a ac Pro. Process running on one core. 15

16 Spectral PIC: ore results for FODO channel umerical emittance growth in a FODO channel after cells. Spectral PIC with =1000. Higher intensity. o sc μ0 = 60, Δμ = 10 o μ0 = 60, Δ μ = 15 o sc o = 64 Artificial collisions? Artificial collisions? (Spectral resolution) Because symplectic PIC solves a macro-particle model, collisions should be present and the emittance should growth in AG focusing according to (still to be shown) : D T m B = ν 1 dε 1 ( T Ty) = Bν ε dt T T 1 / macro-particle number y 16

17 Conclusions and Outloo umerical emittance growth is a concern for long-term PIC particle tracing simulations with space charge in synchrotrons. Two sources of numerical emittance growth: 1) Artificial collisions (or numerical IBS ): Emittance growth in AG focusing ) Stochastic or grid heating ( inelastic collisions ) -> dominant in conventional PIC T Collision rate: B ν D > ν (diffusion, not balanced by friction -> heating) m Counter measures to decrease the growth rate: Larger, particle shapes, filters,. Symplectic PIC: + Still collisional (we don t solve the Vlasov-Poisson equations)! + oise/collisions do not cause emittance growth -> Bounded emittance - Usually slower than conventional PIC. T B D ν m To do: -> optimized implementations, e.g. using a properly chosen S() one a grid. -> chec other benchmars: Resonances, Landau damping and echoes,.. -> compare with other approaches, lie symplectic Vlasov-Poisson solvers 17