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/ 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 http://arxiv.org/abs/1305.7375v2 Details from brendan.godfrey@ieee.orgbrendan.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 6/19/20132

3. Generalized PSATD Algorithm 6/18/20133

4. 2-D High-γ Dispersion Relation 6/18/20134

5. PSATD Normal Modes 6/18/2013 5

6. Full Dispersion Relation Growth Rates 6/19/2013 6 0 +1 +1

7. 6/18/20137

8. 8 Red=1 Purple=0

9. Linear Interpolation, No Filter, γ=130 6/18/20139

10. Cubic Interpolation, Filter, γ=130 6/18/201310 Numerical Cherenkov instability largely eliminated Option (c) residual growth a finite-γ effect

11. Option (c), Cubic Interpolation, Variable Width Sharp Filter, γ=130 6/18/201311

12. WARP Simulations Confirm Numerical Cherenkov Instability Suppression 6/18/201312 Numerical energy growth in 3 cm, γ=13 LPA segment FDTD-CK simulation results included for comparison

13. Analysis Also Valid at Low γ 6/18/201313 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) 6/18/201314

15. First Attempt Partly Successful 6/18/201315

16. Linear Interpolation, No Filter, γ=130 6/18/201316

17. Cubic Interpolation, Sharp Filter, γ=130 6/18/201317

18. New, but Weak, Instability Occurs 6/18/2013 18

19. Next Steps 6/18/201319

20. BACKUP 6/18/201320

21. Analytical Approximations Available 6/18/201321

22. 6/18/201322

23. Cubic Interpolation, Filter, γ=130 6/18/201323