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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. - PowerPoint PPT Presentation
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Numerical Stability of the Pseudo-Spectral EM PIC Algorithm*
Brendan Godfrey, IREAP, U MarylandJean-Luc Vay, Accelerator & Fusion Research, LBNL
Irving Haber, IREAP, U Maryland
LBNL, 19 June 2013
*Research supported in part by US Dept of Energy
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 [email protected]
*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
6/19/2013 2
Generalized PSATD Algorithm
with , , • - free parameters• J assumed to conserve charge– e.g., Esirkepov algorithm or standard current correction
6/18/2013 3
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
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PSATD Normal Modes
Normal ModesParameters
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EM modes fold over when
Full Dispersion Relation Growth Rates
• Peak growth rates at resonances– dominates for – dominates otherwise
• Parameters– Linear interpolation
6/19/2013 6
Numerical Cherenkov instability dominant resonances typically lie at large
0
+1 -1
+1
-1
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,
6/18/2013 7
In general, necessary to avoid new instabilities
, and Digital Filter Plots
6/18/2013 8
• Evaluated for • Ten-pass bilinear filter shown for comparison
, filter transverse current components only
Red=1Purple=0
Linear Interpolation, No Filter, γ=130
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• Option (c) suppresses instability only• Option (d) – conventional current deposition and
correction – included for comparison
Cubic Interpolation, Filter, γ=130
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• Numerical Cherenkov instability largely eliminated• Option (c) residual growth a finite-γ effect
Option (c), Cubic Interpolation,Variable Width Sharp Filter, γ=130
6/18/2013 11
• for , α values in legend• Option (c) filter from last slide kept as baseline
WARP Simulations Confirm Numerical Cherenkov Instability Suppression
6/18/2013 12
• Numerical energy growth in 3 cm, γ=13 LPA segment• FDTD-CK simulation results included for comparison
Analysis Also Valid at Low γ
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• Performed to identify errors not visible at high γ• Electrostatic numerical instability dominates at low γ• Option (b) used
PSATD with Potentials(Use for Pushing Canonical Momenta)
• Gauge invariance: One of four potentials {, } can be specified arbitrarily
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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
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Linear Interpolation, No Filter, γ=130
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• Growth reduced by ¼ at small , by ½ otherwise• Options defined as before, but details of (c) differ
Cubic Interpolation, Sharp Filter, γ=130
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• for , • New instability dominates (but vanishes for )
New, but Weak, Instability Occurs
• Low growth rate• Small k range• Bad location
• Also occurs in PSTD• Parameters– Option (b)– Sharp filter,
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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/
6/18/2013 19
BACKUP
6/18/2013 20
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
6/18/2013 21
, and Digital Filter Plots
6/18/2013 22
• at on left, at on right• Ten-pass bilinear filter shown for comparison• Evaluated for
, filter transverse current components only
Cubic Interpolation, Filter, γ=130
6/18/2013 23
• Growth infinitesimal for • New instability increases option (c) growth at larger ,
but still very small