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Accuracy and Performance Upgrade of PIC and Hybrid Space-Charge Solvers

Accuracy and Performance Upgrade of PIC and Hybrid Space-Charge Solvers. Leonid Vorobiev Muons, Inc., Batavia, IL 60510, USA. ABSTRACT

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Accuracy and Performance Upgrade of PIC and Hybrid Space-Charge Solvers

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  1. Accuracy and Performance Upgrade of PIC and Hybrid Space-Charge Solvers Leonid Vorobiev Muons, Inc., Batavia, IL 60510, USA 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. SPACE CHARGE 2013 16-19 April, CERN L.G.Vorobiev

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

  3. L.G.Vorobiev Some History F.Schmidt APC Seminar at Fermilab, November 2012 H-DETUNING PTC-ORBIT Fake Tune Evolution PTC-ORBIT 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

  4. L.G.Vorobiev Outline PIC Formulation. Clouds & Grid Functions. Errors & Accuracy Upgrades. Limitations. Hybrid Space Charge Solvers. SC Templates. Back to PIC - Split operator revisit. Hierarchy of SC Templates Discussion. Conclusion. Credits. References. SC Solvers vs Applications.

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

  6. L.G.Vorobiev 1.A Grid Density. Known-Unknown? IFSC 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] IFSC densityis approximate, - a starting point in SC errors

  7. L.G.Vorobiev 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. L.G.Vorobiev 1.A Grid Density. Num. Noise Movie Contour lines (left), 3D plot (right). Noise – suppressed. What about Accuracy? 100K Part. 256x256

  9. L.G.Vorobiev 1.A SC Density. Grid Dim & Npart More part. More grid dimensions  Slow Down Vicious circle 1M, 256x256 4M, 256x256 4M, 512x512

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

  11. 1.B Grid Potential and Fields Num. Noise concealed, but persists in dynamics L.G.Vorobiev

  12. L.G.Vorobiev 1.B Field Errors Suppression Field Ex,y errors, depending on SC Clouds What is?

  13. L.G.Vorobiev 1.B Field Errors, Snapshots

  14. 1.B Field Errors. Choice of Field errors, depending on SC Clouds and L.G.Vorobiev

  15. L.G.Vorobiev 1.B Field Errors. Choice of min. But what is ? Simplest benchmarking: Field errors, depending on SC Cloud shape/size

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

  17. L.G.Vorobiev 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. L.G.Vorobiev 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 Green’s function: Ok • Complex Boundary 3D, Sheet Beam (Photo-Injector) Ok • Hybrid SC Solvers

  19. L.G.Vorobiev 2. Hybrid SC Solvers. Outline. Hybrid Space Charge Solvers = Green’s function in Integral form with Tricks The Best onesfor 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: • Space charge templates. Instead of integrated GF, use Library of EXACTSC potential & fields within boundaries • From SC Templates to PIC: Split Operator Revisit • Hierarchy of Templates

  20. L.G.Vorobiev 3.A Hybrid Solvers, SC Templates Disk-Template, Library Ring-Template = Superposition 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 Ref. Vorobiev and York [9]

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

  22. L.G.Vorobiev 3.A Template Solvers, Machinery Beam, Cond.Pipe Template Pots Template Fields Template uzz Template Pots Free space (green) Cond. pipe (blue) Image Dens (red) Beam Pot Beam Field Beam uzz SC Templates - Machinery

  23. L.G.Vorobiev 3.B Hybrid Solvers. From Templates Back to PICs 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) BEAM NOT A ROD E.G. FNAL Booster, MI,… x84,x588 Acceleration with transition crossing Squeeze Head  Tail

  24. L.G.Vorobiev 3.B Hybrid Solvers, Split-Operator Revisit Sub-3D Ref: Vorobiev and York, Phys. Rev. ST Accel. Beams 3, 114201 (2000)

  25. L.G.Vorobiev 3.B Hierachy of Space Charge Templates SC Templates - Flowchart

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

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

  29. L.G.Vorobiev 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. L.G.Vorobiev 5. Conclusion. Credits. 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: Charles Ankenbrandt, Thomas Roberts, Jeffrey Holmes Richard York, Pavel Zenkevich AND Rolland Johnson,Elias Metral and Frank Schmidt

  31. L.G.Vorobiev 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\ Green’s 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 “Green’s 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; PAC’1999, pp. 2781-2783; Springer: http://link.springer.com/chapter/10.1007%2F3-540-47789-6_33 PAC’2001 pp. 3075-3078; EPAC 2002, pp. 1679-1681; PAC’2003 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. L.G.Vorobiev 7. SC Solvers vs Applications Applications to Multi-pass systems • Rings: Multi-Turn Injection • + Acceleration • Storage Rings, Colliders Single-pass systems • (Photo-) Injectors, Electron Optics • Linacs: LBT, Acceleration • Transport, Final Focusing  Yes / Maybe / Not  ?(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)? …

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