Some numerical techniques developed in the Heavy-Ion Fusion program

1. Some numerical techniques developed in the Heavy-Ion Fusion program J.-L. Vay - Lawrence Berkeley National Laboratory Collaborators: • A. Friedman, D.P. Grote - Lawrence Livermore National Laboratory • J.-C. Adam, A. Héron - CPHT, Ecole Polytechnique, France • P. Colella, P. McCorquodale, D. Serafini - Lawrence Berkeley National Laboratory • The Heavy-Ion Fusion program has developed, and continues to work on, numerical techniques that have broad applicability: • Absorbing Boundary Conditions (ABC) • Adaptive Mesh Refinement (AMR) for Particle-In-Cell (PIC) • Advanced Vlasov methods (moving grid,AMR) • Cut-cell boundaries Short-Pulse Laser Matter Computational Workshop Pleasanton, California - August 25-27, 2004

2. Absorbing Boundary Condition: Extended PML

3. Maxwell Extended PML Split Maxwell Berenger PML If and => Z=Z0: no reflection. If with u=(x,y) => Z=Z0: no reflection. Extended Perfectly Matched Layer • Principles of PML: • Field vanishes in layer surrounding domain. • Layer medium impedance Z matches vacuum’s Z0

4. Berenger PML - matched coefficients Plane wave analysis of PML and Extended PML Berenger PML t=2p/w~20dx/c Extended PML • matching condition on coefficients in PML layer improves absorption • Extended PML another overall improvement

5. Extended PML implemented in EM PIC code Emi2d Extended PML

6. Adaptive Mesh Refinement for Particle-In-Cell

7. End-to-end modeling of a Heavy Ion Fusion driver km mm m challenging because length scales span a wide range: mm to km(s)

8. 3D AMR simulation of an explosion (microseconds after ignition) AMR concentrates the resolution around the edge which contains the most interesting scientific features. The Adaptive-Mesh-Refinement (AMR) method • addresses the issue of wide range of space scales • well established method in fluid calculations • potential issues with PIC at interface • spurious self-force on macro-particles • violation of Gauss’ Law • spurious reflection of short wavelengths with amplification

9. 3D WARP simulation of High-Current Experiment (HCX) Modeling of source is critical since it determines initial shape of beam WARP simulations show that a fairly high resolution is needed to reach convergence

10. R (m) zoom Z (m) Z (m) Z (m) Refinement of gradients: emitting area, beam edge and front. Example of AMR calculation with WARPrz: speedup ~10.5 R (m)

11. 3D WARP simulation of HCX shows beam head scrapping Rise-time t = 800 ns beam head particle loss < 0.1% x (m) z (m) Rise-time t = 400 ns zero beam head particle loss x (m) • Simulations show: head cleaner with shorter rise-time • Question: what is the optimal rise-time? z (m)

12. di virtual surface current time Vi irregular patch in di + AMR following front “L-T” waveform N = 160 Dt = 1ns d = 0.4m AMR ratio = 16 I (A) Time (s) Time (s) 1D time-dependent modeling of ion diode Emitter Collector d V V=0 irregular patch in di Ns = 200 dx0/Dx~10-5! Insufficient resolution of beam front => AMR patch Careful analysis shows that di too large by >104 => irregular patch Time (s) MR patch suppresses long wavelength oscillation - AMR patch suppresses front peak

13. STS500 experiment X (m) Z (m) Application to three dimensions • Specialized 1-D patch implemented in 3-D injection routine (2-D array) • Extension Lampel-Tiefenback technique to 3-D implemented in WARP • predicts a voltage waveform which extracts a nearly flat current at emitter • Run with MR predicts very sharp risetime (not square due to erosion) • Without MR, WARP predicts overshoot “Optimized” Voltage Current at Z=0.62m V (kV) T (ms)

14. MR patch Current history (Z=0.62m) Current history (Z=0.62m) MR off MR on MR patch key in simulation of STS500 Experiment • Mesh Refinement essential to recover experimental results • Ratio of smaller mesh to main grid mesh ~ 1/1000

15. Patch 2s=28/k0 fine F Extended PML core C Laser beam Outside patch: F = FM coarse Inside patch: F = FM-FC+FF coarse M Mesh refinement by substitution 10nc, 10keV l=1mm, 1020W.cm-2 (Posc/mec~8,83) Applied to Laser-plasma interaction in the context of fast ignition New MR method implemented in EM PIC code Emi2d

16. Illustration of instability in 1-D EM tests o: E, x:B Space only Space+Time Most MR schemes relying on interpolations are potentially unstable.

17. same results except for small residual incident laser outside region of interest • no instability nor spurious wave reflection observed at patch border Comparison patch on/off very encouraging MR off MR on

18. Effort to develop AMR library for PIC at LBNL • Researchers from AFRD (PIC) and ANAG (AMR-Phil Colella’s group) collaborate to provide a library of tools that will give AMR capability to existing PIC codes (on serial and parallel computers) • The base is the existing ANAG’s AMR library Chombo • The way it works • WARP is test PIC code but library will be usable by any PIC code

19. Example of WARP-Chombo injector field calculation • Chombo can handle very complex grid hierarchy • Electrostatic solver implemented, electromagnetic solver planned

20. References Extended PML J.-L. Vay, “Asymmetric Perfectly Matched Layer for the Absorption of Waves”, J. Comp. Physics183, 367-399 (2002) AMR-PIC Vay JL., Colella P., Kwan JW., McCorquodale P., Serafini DB., Friedman A., Grote DP., Westenskow G., Adam JC. ,Heron A., Haber I., “Application of adaptive mesh refinement to particle-in-cell simulations of plasmas and beams”, Physics of Plasmas, 11(5), 2928-2934, 2004