1. Modeling laser-plasma interaction with the direct implicit PIC method 7 th Direct Drive and Fast Ignition Workshop, Prague, 3-6 May 2009 M. Drouin a, L. Gremillet a, J.-C. Adam b and A. Héron b a CEA, DAM, DIF, Bruyères-le-Châtel, F-91297 Arpajon, France b CPhT, UMR 7644, Ecole Polytechnique, 91128 Palaiseau, France

2. Introduction  Large space- and time-scale particle-in-cell simulations of a high intensity laser (I  > 10 18 Wcm -2 ) interacting with a solid-density target are crucial for many applications (fast ignition, isochoric heating, ion acceleration, ps X-  light sources …). Yet the standard PIC method is based on an explicit scheme which suffers from strong stability constraints.  We therefore propose to solve the Vlasov-Maxwell system by an implicit method 1,2 adapted to the relativistic regime and the propagation of light waves. Such a scheme could provide an increased numerical stability for large spatial and temporal step sizes, when also providing satisfactory energy conservation. 1 D. W. Hewett and A. B. Langdon, J. Comput. Phys. 72, 121-155 (1987) 2 D. Welch, D. Rose, B. Oliver, and R. Clark, Nucl. Instrum. Methods Phys. Res. A 464, 134-139 (2001)

3. Summary  Basic principles of the direct implicit method  Results and benchmarks Comparison between implicit and explicit discretizations Design of a predictor-corrector scheme Adjustable damping and electromagnetic propagation into vacuum Electrostatic dispersion relation of a warm plasma including  x and  t Plasma expansion into vacuum Laser-plasma interaction in the overcritical regime

4. A comparison between implicit and explicit discretizations Explicit 1 methodDirect implicit 2 method 2 D. W. Hewett and A. B. Langdon, J. Comput. Phys. 72, 121 (1987) 1 C. K. Birdsall and A. B. Langdon, Plasma physics via computer simulation (1985)  Relativistic Lorentz’ equations  Maxwell’s equations (Yee’s scheme)  Maxwell’s equations  Relativistic Lorentz’ equations  Properties Pusher stability Maxwell stability (CFL) (harmonic force) (plasma wave)  Properties Strong damping of high frequency modes Stability in a broader (  x,  t) range No CFL constraint on the electromagnetic solver in vacuum

5. Design of a predictor-corrector scheme  Substituting the associated currents into Maxwell’s equations, we get 1 P. Concus and G.H. Golub SIAM Journal on Numerical Analysis 10, 1103-1120 (1973)  Eventually we solve the wave equation using an iterative method 1  Correction terms are functions of the future fields : with Relativistic susceptibilities for a particle i are given by  Predicted positions and momenta are functions of known fields :

6. Adjustable damping and electromagnetic propagation into vacuum 1 A. Friedman, J. Comput. Phys. 90, 292-312 (1990) θ f = 0 θ f = 1 θ f = 0  Limit cases θ f = 0  θ f = 1   We have adapted Friedman’s 1 scheme to the discretization of Maxwell’s equations :  Numerical example k 0  x = 0.2 ; k 0  y = 0.8 ; ω 0  t = 0.2 1025×4 mesh θ f = 1 (damping) θ f = 0 (no damping)  Original implicit scheme is strongly dissipative for both plasma and light waves

7. Electrostatic dispersion relation including finite space and time discretizations  Assuming an infinite 1d Maxwellian plasma we establish the electrostatic dispersion relation coupling the complex frequency  and the wave number k :  Aliasing may produce instability or damping  In general the damping/growth rate is a function of  p  t,  x/ D, θ f and the order of the weight function denotes the Fried & Conte function and Implicit part Explicit (leapfrog)

8. Damping/Stabilizing role of the time step ω p Δt >1 and of the shape function Order 1Order 2 ω p Δt = 1 Γ max =+1.8×10 -2 k max  x=2.54 Γ max =+3×10 -3 k max  x=2.41 ω p Δt = 2 Γ max =+10 -2 k max  x=2.54 Γ max =-5.2×10 -3 k max  x=2.47 ω p Δt = 5 Γ max =-1.1×10 -2 k max  x=2.56 Γ max =-2.7×10 -2 k max  x=2.49 1d Maxwellian plasma with   x/λ D ~ 30  damping parameter θ = 1. ΔtΔt

9. Comparison with simulations of a freely expanding 1D Maxwellian plasma  Maxwellian plasma expansion in vacuum : L plasma = 18.84 c/ω 0 m i /m e =900 ; T e =T i =1 keV  x =  y = 0.2 (c/ω 0 ) n e = 44 n c,  x/λ D ~ 30 60 particles/mesh 300×4 meshes  Total plasma energy variation per time step : Order 1Order 2 ω p Δt = 11.7×10 -3 4.6×10 -4 ω p Δt = 21.2×10 -3 3.2×10 -4 ω p Δt = 53×10 -4 0 ω p Δt=1 ω p Δt=2 ω p Δt=5 Order 2 ω p Δt=1 ω p Δt=2 ω p Δt=5 Order 1 Ecin_i Ec,tot Ecin_e

10. Plasma expansion into vacuum: comparison with explicit simulation Explicit relativistic (Calder) ω p Δt = 0.1, (ω p /c)  x = 0.2 so  x/λ D # 1.4 Kinetic energies e- i  E/E 0 ≈ +1% Implicit relativistic ω p Δt = 2, (ω p /c)  x = 2 so  x/λ D # 14 Kinetic energies e- i  E/E 0 ≈ -2.8% 2dx3dv Maxwellian plasma T e = 10 keV, T i = 0.5 keV n e = n i = 100 n c Periodic boundary conditions along y Linear weight function 600 × 10 3 particles (explicit) and 60 × 10 3 (implicit) Explicit : 1h16 × 4 proc Implicit : 12 min

11. Laser-plasma interaction in overcritical regime  2dx3dv explicit simulation (Calder)  t = 0.05 ω 0 -1  x =  y = 0.08 (c/ω 0 ) 3 rd order weight factor 160 particles/mesh  2dx3dv implicit simulation  t = 0.3 ω 0 -1 (beyond CFL)  x =  y = 0.1 (c/ω 0 ) ω p Δt/(  x/λ D ) ~ 0.13 2 nd order weight factor 40 particles/mesh 1 200 ω 0 > ω p ω 0 < ω p ω p Δt < ω 0 Δt ≤ 1ω p Δt ≥ 1 Dense plasma T e = T i = 1 keV Slightly dispersive scheme θ f =0.1 Laser I = 10 19 W/cm 2 x Conservative scheme θ f = 0 ne/ncne/nc 1  m2  m

12. Evolution of kinetic energies and phase spaces Explicit relativistic Implicit relativistic Electronic (left) and ionic (right) phase spaces (x, p x ) 4.8% energy balance (heating) -11.2% energy balance (cooling) 64×4.6h ≈ 290h 1×27.5h

13. Hot electron generation and distribution Explicit relativistic Implicit relativistic Hot electron production, bunched acceleration and transport through the dense slab are well reproduced Energy distributions Explicit Implicit

14. Conclusions and prospects  Validation of the relativistic direct implicit method with adjustable damping. Application to relativistic laser-plasma interaction.  Good energy conservation properties of the implicit scheme 1. Benefit of high order weight functions 2,3,4 Future work  Introduction of binary relativistic collisions in order to describe dense plasmas.  Parallelisation to study more realistic 2D/3D configurations. 1 B. I. Cohen et al., J. Comput. Phys. 81, 151 (1989) 2 S. D. Baton et al., Phys. Plasmas 15, 042706 (2008) 3 R. Nuter et al., soumis à JAP (2008) 4 M. Drouin et al., in preparation (2009)

15. Long irradiation simulations : hot electron generation Explicit relativisticImplicit relativistic

16. Long irradiation simulations : Quasistatic magnetic field generation Implicit relativistic

17. About the linearisation of 1/γ n  ELIXIRS formulation of the velocity correction term, obtained by strict linearisation of the Lorentz’ equations assuming :  LSP formulation of the velocity correction term :  where the exact and approximated Lorentz’ factors are defined as

18. Over-critical laser plasma interaction (1/2)  High intensity laser interaction with an over- critical plasma slab preceded by a plasma ramp : n e max = n i max = 200 n c 2 nd order weight factor  x =  y = 0.1 (c/ω 0 )  x/λ D ~ 32 2000 particules/maille 2048 × 4 cells 1 200 ω 0 > ω p ω 0 < ω p ω p Δt < ω 0 Δt ≤ 1ω p Δt ≥ 1 Dense plasma T e = T i = 1 keV Slightly dispersive scheme θ f = 0.05 Laser I = 10 19 W/cm 2 x Conservative scheme θ f = 0 ne/ncne/nc 1  m3  m  2dx3dv explicit simulation (Calder)  t = 0.05 ω 0 -1  2dx3dv implicit simulation  t = 0.141 ω 0 -1

19. Over-critical laser plasma interaction (2/2) Explicit relativistic (Calder) Implicit relativistic Total energies Electronic (left) and ionic (right) phase spaces (x, p x ) Total energies 17h30 >48h