1. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 1 Accuracy and Performance Upgrade of PIC and Hybrid Space-Charge Solvers Leonid Vorobiev Muons, Inc., Batavia, IL 60510, USA SPACE CHARGE 2013 16-19 April, CERN ABSTRACT For the high luminosity LHC (HL-LHC), space charge study requires high quality modeling. In this talk the sources of numerical errors in space charge algorithms are analyzed, and their mitigation is suggested for both conventional space charge grid solvers (based on PIC formalism) and hybrid solvers. The implementation of these improvements may significantly increase the accuracy, speed and physical validity of numerical results and contribute to achieve the required beam parameters in the LHC accelerator complex.

2. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 2 Overview General SC: Grid(PIC), Hybrid Complex boundaries, Image Forces, Wakes, Beam Loading, Halo, Clusters... Works fine for many applications at Macro-level High-Current Beam Simulation = Tracker & SC Solver Symplectic maps Integrators, high order Tracking engines Tracking engines = & SC Zero/Non-Zero Frozen SC Free space, Pre-assigned Distribution: KV, Gaussian,…, A starting point EXACT, Not GeneralAPPROXIMATE, General ~EXACT Recent findings: Frank Schmidt, et al. Micro-scale effects ARE persistent. E.G.: ORBIT code J.Galambos, J.Holmes, D.Olsen, A.Luccio and J.Beebe-Wang, 1999.

3. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 3 Some History Micro-scale unphysical effects: H-detuning (left), fake tune Evolutions, artificial dipole oscillations,… May jeopardize the validity of beam modeling WANTED: Consistency between ~Exact Trackers and approximate SC Solvers Search for remedies Fake Tune Evolution PTC-ORBITH-DETUNING PTC-ORBIT F.Schmidt APC Seminar at Fermilab, November 2012

4. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 4 Outline 1. PIC Formulation. Clouds & Grid Functions. Errors & Accuracy Upgrades. Limitations. 2. Hybrid Space Charge Solvers. 3. SC Templates. Back to PIC - Split operator revisit. Hierarchy of SC Templates 4. Discussion. 5. Conclusion. Credits. 6. References. 7. SC Solvers vs Applications.

5. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 5 1. PIC Formulation The Greens function in Differential form, for electrostatic approximation, leads to STEPS: A.Build a Grid. Charge density on the grid. B.Potential/Fields on the grid ( FFT-series, Multi-Grid, Iterations with accel., Boundaries/Symmetries. C.Interpolation between nodes. Refs.: Birdsall and Langdon [1], Hockney and Eastwood [2] Poisson equation:

6. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 6 1.A Grid Density. Known-Unknown? IF SC density is known (Gaussian,…exact) (Multi-) Grid solvers may approach a solution with machine accuracy, exact SC Fields Refs. www.nr.com [3], Saraniti [4] IF SC density is approximate, - a starting point in SC errors

7. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 7 1.A Grid Density. Clouds. Contour lines (left), 3D plot (right). Simplest Cloud. Strong Numerical noise PIC Clouds: Size, Shape, Ramp Redistribution of SC over the grid From: Ref. [5], Vorobiev and Hirata.

8. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 8 1.A Grid Density. Num. Noise Movie Contour lines (left), 3D plot (right). Noise – suppressed. What about Accuracy? 100K Part. 256x256

9. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 9 1.A SC Density. Grid Dim & N part 1M, 256x256 4M, 256x256 4M, 512x512 More part. More grid dimensions Slow Down Vicious circle

10. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 10 1.A Grid Density. Num. Noise Movie Contour lines (left), 3D plot (right). Noise – suppressed. What about Accuracy? 4M Part. 512x512

11. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 11 1.B Grid Potential and Fields Num. Noise concealed, but persists in dynamics

12. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 12 1.B Field Errors Suppression Field E x,y errors, depending on SC Clouds What is ?

13. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 13 1.B Field Errors, Snapshots

14. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 14 1.B Field Errors. Choice of Field errors, depending on SC Clouds and

15. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 15 Field errors, depending on SC Cloud shape/size min. But what is ? Simplest benchmarking: 1.B Field Errors. Choice of

16. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 16 1.B Clusters All this works similarly for Symmetries, Boundary constraints, as well as for clustered beams Contour lines, 3D shape, Potential (top), Fields (bottom)

17. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 17 1.C Interpolation. SC Fields are solved on the Grid Particles are in between Grid nodes: Bi-Tri-Piecwise Interpolation -Wrong Bi-Tri-Linear Interpolation - Insufficient Bi-Tri-Cubic Interpolation: Splines - Best Ref. Vorobiev and Hirata [5]

18. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 18 1. Grid Solvers. Limitations PIC Grid Solvers: solid/clustered beams, Ok Filaments, Sheet beams, Halo, Stray particles not Ok Mesh refinement, Density errors, irregular Clouds, use more particles Errors due to interpolation on irregular grid Slowing down speed… Aside from PICs: Particle-Core, Envelope (linear) & non-linear model ? Integral representation of the Greens function: Ok Complex Boundary 3D, Sheet Beam (Photo-Injector) Ok Hybrid SC Solvers

19. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 19 2. Hybrid SC Solvers. Outline. Hybrid Space Charge Solvers = Greens function in Integral form with Tricks The Best ones for photo-injectors, el.optics, complex boundaries: VERY accurate, SLOW Refs: Roger Harrington [6], Miklosh Szilagyi [7], Valentin Ivanov [8], In part: Fikera, Quang, Ryan, Kapin, Hess, Krassilnikov, Vorobiev This Presentation: A. Space charge templates. Instead of integrated GF, use Library of EXACT SC potential & fields within boundaries Split Operator Revisit B. From SC Templates to PIC: Split Operator Revisit C. Hierarchy of Templates

20. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 20 Disk-Template, Library Ring-Template = Superposition of positive (BLUE) & negative (RED) derived from Disc-Template of positive (BLUE) & negative (RED) derived from Disc-Templates 3D beam (non-elliptical!) Within conducting Boundary (not shown) 3D Templates within the same boundary NO GRID 3D Poisson Equation, INSTEAD: Convolve Template Potentials/Fields 3.A Hybrid Solvers, SC Templates Ref. Vorobiev and York [9]

21. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 21 Library of Templates Pre-Assigned densities Ring Disc Template General Halo Hollow Beam 3.A Hybrid Solvers, SC Templates Private Communications: A. Friedman, D. Grote, I. Hofmann, M. Reiser, J. Struckmeier and R. York

22. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 22 3.A Template Solvers, Machinery SC Templates - Machinery Template Pots Free space (green) Cond. pipe (blue) Image Dens (red) Beam, Cond.Pipe Template Pots Template Fields Template u zz Beam Pot Beam Field Beam u zz

23. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 23 3.B Hybrid Solvers. From Templates Back to PICs BEAM NOT A ROD E.G. FNAL Booster, MI,… x84,x588 Acceleration with transition crossing Squeeze Head Tail 2.5D – series of 2D solutions, as if BEAM = ROD BEAM=ROD model - Good for PSR, SNS,… 2D beam, 1 long bunch 80% of circumference)

24. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 24 3.B Hybrid Solvers, Split-Operator Revisit Ref: Vorobiev and York, Phys. Rev. ST Accel. Beams 3, 114201 (2000) Sub-3D

25. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 25 3.B Hierachy of Space Charge Templates SC Templates - Flowchart

26. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 26 4. Discussion. Existing SC Classes. Project X, Project X, Main Injector, H- Multiturn Injection Transverse Painting (x,x),(y,y), Longitudinal Painting FNAL Booster Transition Crossing: RF cavities - narrow SC issues Ref. Vorobiev [11-12]

27. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 27 Mu2e Exp. In Fermilab: Extinction interval for mu2e detector: NO STRAY PROTONS Long. dynamics w/o (left), with (cntr) beam loading. Histograms (right) 4. Discussion. Existing SC Classes Ref. Balbekov and Vorobiev [10]

28. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 28 4. Discussion: Done,…, TBD Accuracy upgrades SC Density. Clouds. Size. Shape. + Interpolation. Splines… + 3D SC Templates for E z + Revisit Split Operator, Sub-3D + 2D Templates, Envelope & Images + Halo TBD Memory, Speed No Grid Much lesser part Lesser Memory demands due to parameterization Beam Loading & Wakes, 1++ Kicks/turn? TBD Existing SC modules: Multi-turn H- injection (Project-X), Beam Loading (mu2e), Long. Dynamics (Transition Xing, Head-Tail… On the way to full Synchro-Betatron dynamics studies

29. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 29 4. Discussion. HL-LHC Computational Needs HL-LHC high intensity mode of operation Linacs Series of Synchrotrons LHC Linacs, Multiturn Injection, Acceleration, Trans. Crossing, Circulation Tune shift, Image forces, Wakes, B. Loading Halo, Losses… Wanted: SC Solvers – consistent to Symplectic Trackers and Integrators SC Performance - Overall Accuracy & Speed Upgrade, Reasonable Memory Capacities, Physical Validity

30. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 30 5. Conclusion. Credits. Charles Ankenbrandt, Thomas Roberts, Jeffrey Holmes Richard York, Pavel Zenkevich AND Rolland Johnson, Elias Metral and Frank Schmidt Optional Space Charge Solvers in existing code Cross-checking between codes, Stress/Regression Tests Keep development of different codes by different authors – MUST GOAL: reliable, insightful modeling for HL-LHC Credits to:

31. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 31 6. References PIC 1. C.K.Birdsall and A.B. Langdon (1985). Plasma Physics via Computer Simulation. McGraw-Hill. 2. R.W.Hockney and J.W. Eastwood (1988). Computer Simulation Using Particles. CRC Press. Multi-Grid Poisson Solvers 3.W.H. Press, S.A.Teukolsky, W.T.Vetterling, B.P.Flannery, Numerical recipes, www.nr.com. 4. M.Saraniti, et al. IEEE Transactions on CAD, Volume: 15 Issue: 2, Page(s): 141-150 (1996). Symplectic SC solver 5. Vorobiev and Hirata, Report KEK 95-12; http://ccdb5fs.kek.jp/cgi-bin/img/allpdf?199524012\ Greens Function 6. R.Harrington, Field Computation by Moment Methods, Macmillan, New York (1968) 7. M.Szilagyi, Electron and Ion Optics, Plenum, New York (1988) 8. V.Ivanov Greens Function Techniques in Forming Intense Beams, (1989, 2009), Int. J.Mod.Phys A, Vol.24, No. 5, 869–878,…In part: Quang, et al. (2006), Kapin (2002), Hess, et al.(2007), Vorobiev (1999). SC Templates: 9. Vorobiev and York: Phys.Rev. STAB 3, 114201 (2000); MSUCL-Report 1117, 1998; PAC1999, pp. 2781-2783; Springer: http://link.springer.com/chapter/10.1007%2F3-540-47789-6_33 PAC2001 pp. 3075-3078; EPAC 2002, pp. 1679-1681; PAC2003 pp. 3533-3535; Fermilab Report 08-236, http://lss.fnal.gov/archive/2008/pub/fermilab-pub-08-236-apc.pdf BL + Transition crossing: 10. Balbekov and Vorobiev, http://beamdocs.fnal.gov/AD-public/DocDB/ShowDocument?docid=4070 11. Vorobiev, https://indico.fnal.gov/conferenceDisplay.py?confId=4497 http://beamdocs.fnal.gov/AD-public/DocDB/ShowDocument?docid=4134 12. Vorobiev, http://beamdocs.fnal.gov/AD-public/DocDB/ShowDocument?docid=4258

32. Muons, Inc. CERN, Space Charge 2013, April 17L.G.Vorobiev 32 7. SC Solvers vs Applications Applications to Single-pass systems (Photo-) Injectors, Electron Optics Linacs: LBT, Acceleration Transport, Final Focusing Multi-pass systems Rings: Multi-Turn Injection + Acceleration Storage Rings, Colliders ?(maybe/no) Symplecticity (maybe/yes)? ?(maybe/yes) Complex Boundary (no)? ?(maybe/yes) Multi-step SC (no/maybe UMd ERing )? ?(maybe/no) SC Kicks (yes)? ?(maybe) Beam Loading, Wakes (yes)? … Yes / Maybe / Not Yes / Maybe / Not