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Numerical Stability of the Pseudo-Spectral EM PIC Algorithm*

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.

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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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