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Numerical Stability of the Pseudo-Spectral EM PIC Algorithm*

This presentation discusses the enhancements made to the Pseudo-Spectral "Analytical" Time Domain (PSATD) Algorithm for relativistic beam simulations, achieving a significant reduction in numerical Cherenkov instability. Key findings highlight the strategies implemented to suppress instability through various filtering options, alongside performance validation in WARP simulations. The research, conducted by Brendan Godfrey (U Maryland), Jean-Luc Vay (LBNL), and Irving Haber (U Maryland), showcases future steps to refine the algorithm and continue exploring its capabilities. Access additional resources and software at the provided links.

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Numerical Stability of the Pseudo-Spectral EM PIC Algorithm*

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  1. Numerical Stability of the Pseudo-Spectral EM PIC Algorithm* Brendan Godfrey, IREAP, U Maryland Jean-Luc Vay, Accelerator & Fusion Research, LBNL Irving Haber, IREAP, U Maryland LBNL, 19 June 2013 *Research supported in part by US Dept of Energy

  2. Pseudo-Spectral “Analytical” Time Domain (PSATD) Algorithm* • Numerical Cherenkov instability largely eliminated for PSATD relativistic beam simulations • Numerical stability software collection at http://hifweb.lbl.gov/public/BLAST/Godfrey/ • Contains this talk, some software; more to come • Algebra omitted from this talk available at http://arxiv.org/abs/1305.7375v2 • Details from brendan.godfrey@ieee.org *Introduced by I. Haber, et. al., Sixth Conf. Num. Sim. Plas. (1970) Expanded upon by J.-L. Vay, et. al., paper 1B-1, PPPS 2013

  3. Generalized PSATD Algorithm with , , • - free parameters • J assumed to conserve charge • e.g., Esirkepov algorithmor standard current correction

  4. 2-D High-γ Dispersion Relation with , • Dispersion relation reduces to in continuum limit (n is density divided by γ) • Beam modes are numerical artifacts, trigger numerical Cherenkov instability • May have other deleterious effects

  5. PSATD Normal Modes Normal Modes Parameters EM modes fold over when

  6. Full Dispersion Relation Growth Rates • Peak growth rates at resonances • dominates for • dominates otherwise • Parameters • Linear interpolation +1 -1 +1 0 -1 Numerical Cherenkov instability dominant resonances typically lie at large

  7. Can Reduce Instability • Option (a) - • Option (b) - • Option (c) - to suppress instability • Choose so that at resonance • Verify as (it does) • Set when it falls outside • Choose any reasonable ‒ here, In general, necessary to avoid new instabilities

  8. , and Digital Filter Plots • Evaluated for • Ten-pass bilinear filter shown for comparison Red=1 Purple=0 , filter transverse current components only

  9. Linear Interpolation, No Filter, γ=130 • Option (c) suppresses instability only • Option (d) – conventional current deposition and correction – included for comparison

  10. Cubic Interpolation, Filter, γ=130 • Numerical Cherenkov instability largely eliminated • Option (c) residual growth a finite-γ effect

  11. Option (c), Cubic Interpolation,Variable Width Sharp Filter, γ=130 • for , α values in legend • Option (c) filter from last slide kept as baseline

  12. WARP Simulations Confirm Numerical Cherenkov Instability Suppression • Numerical energy growth in 3 cm, γ=13 LPA segment • FDTD-CK simulation results included for comparison

  13. Analysis Also Valid at Low γ • Performed to identify errors not visible at high γ • Electrostatic numerical instability dominates at low γ • Option (b) used

  14. PSATD with Potentials(Use for Pushing Canonical Momenta) • Gauge invariance: One of four potentials {,} can be specified arbitrarily

  15. First Attempt Partly Successful • Choose gauge • Resulting high-γ dispersion relation: • Reduces order of spurious beam mode from 2 to 1 • Introduces spurious vacuum mode

  16. Linear Interpolation, No Filter, γ=130 • Growth reduced by ¼ at small , by ½ otherwise • Options defined as before, but details of (c) differ

  17. Cubic Interpolation, Sharp Filter, γ=130 • for , • New instability dominates (but vanishes for )

  18. New, but Weak, Instability Occurs • Low growth rate • Small k range • Bad location • Also occurs in PSTD • Parameters • Option (b) • Sharp filter,

  19. Next Steps • Implement algorithm in WARP, validate results • Obtain Option (c) for finite γ • Explore other PSATD variants • Understand, suppress new instability • Generalize to FDTD • Add more material to http://hifweb.lbl.gov/public/BLAST/Godfrey/

  20. Backup

  21. Analytical Approximations Available • Resonant instability - • Typically so strong it must be filtered digitally • But, not difficult to do • Nonresonant instability - • Not so strong, but often lies at small • Filtering more difficult to accomplish • So, combine filtering with making • Eliminating nonresonant instability the focus of talk

  22. , and Digital Filter Plots • at on left, at on right • Ten-pass bilinear filter shown for comparison • Evaluated for , filter transverse current components only

  23. Cubic Interpolation, Filter, γ=130 • Growth infinitesimal for • New instability increases option (c) growth at larger , but still very small

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