modeling laser-plasma interaction with the direct implicit pic method 7 th direct drive and fast...
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Modeling laser-plasma interaction with the direct implicit PIC method
7th Direct Drive and Fast Ignition Workshop, Prague, 3-6 May 2009
M. Drouina, L. Gremilleta, J.-C. Adamb and A. Héronb
a CEA, DAM, DIF, Bruyères-le-Châtel, F-91297 Arpajon, Franceb CPhT, UMR 7644, Ecole Polytechnique, 91128 Palaiseau, France
Introduction
Large space- and time-scale particle-in-cell simulations of a high intensity laser (I> 1018 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 schemewhich suffers from strong stability constraints.
We therefore propose to solve the Vlasov-Maxwell system by an implicit method1,2 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.
1D. W. Hewett and A. B. Langdon, J. Comput. Phys. 72, 121-155 (1987)
2D. Welch, D. Rose, B. Oliver, and R. Clark, Nucl. Instrum. Methods Phys. Res. A 464, 134-139 (2001)
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
A comparison between implicit and explicit discretizations
Explicit1 method Direct implicit2 method
2D. W. Hewett and A. B. Langdon, J. Comput. Phys. 72, 121 (1987)
1C. 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
Design of a predictor-corrector scheme
Substituting the associated currents into Maxwell’s equations, we get
1P. Concus and G.H. Golub SIAM Journal on Numerical Analysis 10, 1103-1120 (1973)
Eventually we solve the wave equation using an iterative method1
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 :
Adjustable damping and electromagnetic propagation into vacuum
1 A. Friedman, J. Comput. Phys. 90, 292-312 (1990) θf = 0 θf = 1 θf = 0
Limit cases• θf = 0 • θf = 1
We have adapted Friedman’s1 scheme to the discretization of Maxwell’s equations :
Numerical example • k0x = 0.2 ; k0y = 0.8 ; ω0t = 0.2• 1025×4 mesh
θf = 1(damping)
θf = 0
(no damping)
Original implicit scheme is strongly dissipative for both plasma and light waves
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 pt, x/D, θf and the order of the weight function
denotes the Fried & Conte function and
Implicit part Explicit (leapfrog)
Damping/Stabilizing role of the time step ωpΔt >1 and of the shape function
Order 1 Order 2
ωpΔt = 1 Γmax=+1.8×10-2
kmax x=2.54
Γmax=+3×10-3
kmax x=2.41
ωpΔt = 2 Γmax=+10-2
kmax x=2.54
Γmax=-5.2×10-3
kmax x=2.47
ωpΔt = 5 Γmax=-1.1×10-2
kmax x=2.56
Γmax=-2.7×10-2
kmax x=2.49
1d Maxwellian plasma withx/λD ~ 30 damping parameter θ = 1.
Δt
Comparison with simulations of a freely expanding 1D Maxwellian plasma
Maxwellian plasma expansion in vacuum :• Lplasma = 18.84 c/ω0
• mi/me=900 ; Te=Ti=1 keV• x = y = 0.2 (c/ω0)• ne = 44 nc, x/λD ~ 30• 60 particles/mesh• 300×4 meshes
Total plasma energy variation per time step :
Order 1 Order 2
ωpΔt = 1 1.7×10-3 4.6×10-4
ωpΔt = 2 1.2×10-3 3.2×10-4
ωpΔt = 5 3×10-4 0
ωpΔt=1
ωpΔt=2
ωpΔt=5
Order 2
ωpΔt=1
ωpΔt=2
ωpΔt=5
Order 1
Ecin_i
Ec,tot
Ecin_e
Plasma expansion into vacuum: comparison with explicit simulation
Explicit relativistic (Calder)ωpΔt = 0.1, (ωp/c)x = 0.2 so x/λD # 1.4
Kinetic energies
e-
i
E/E0 ≈ +1%
Implicit relativisticωpΔt = 2, (ωp/c)x = 2 so x/λD # 14
Kinetic energies
e-
i
E/E0 ≈ -2.8%
• 2dx3dv •Maxwellian plasma Te = 10 keV, Ti = 0.5 keV • ne = ni = 100 nc • Periodic boundary conditions along y• Linear weight function
• 600×103 particles (explicit) and 60×103 (implicit)
Explicit :1h16 × 4 proc Implicit :12 min
Laser-plasma interaction in overcritical regime
2dx3dv explicit simulation (Calder)• t = 0.05 ω0
-1 • x = y = 0.08 (c/ω0)• 3rd order weight factor• 160 particles/mesh
2dx3dv implicit simulation• t = 0.3 ω0
-1 (beyond CFL)• x = y = 0.1 (c/ω0) ωpΔt/(x/λD) ~ 0.13• 2nd order weight factor• 40 particles/mesh
1
200
ω0 > ωp
ω0 < ωp
ωpΔt < ω0Δt ≤ 1 ωpΔt ≥ 1
Dense plasmaTe = Ti = 1 keV
Slightly dispersive scheme θf =0.1
Laser I = 1019 W/cm2
x
Conservative scheme θf = 0
ne/nc
1 m2 m
Evolution of kinetic energies and phase spaces
Explicit relativistic
Implicit relativistic
Electronic (left) and ionic (right) phase spaces (x, px)
Electronic (left) and ionic (right) phase spaces (x, px)
4.8% energy balance (heating)
-11.2% energy balance (cooling)
64×4.6h ≈ 290h
1×27.5h
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
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 scheme1. Benefit of high order weight functions2,3,4
Future work Introduction of binary relativistic collisions in order to describe dense plasmas.
Parallelisation to study more realistic 2D/3D configurations.
1B. I. Cohen et al., J. Comput. Phys. 81, 151 (1989)
2S. D. Baton et al., Phys. Plasmas 15, 042706 (2008)
3R. Nuter et al., soumis à JAP (2008)
4M. Drouin et al., in preparation (2009)
About the linearisation of 1/γn
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
Over-critical laser plasma interaction (1/2)
High intensity laser interaction with an over-critical plasma slab preceded by a plasma ramp : • ne
max = nimax = 200 nc
• 2nd order weight factor• x = y = 0.1 (c/ω0) x/λD ~ 32• 2000 particules/maille • 2048 × 4 cells
1
200
ω0 > ωp
ω0 < ωp
ωpΔt < ω0Δt ≤ 1 ωpΔt ≥ 1
Dense plasmaTe = Ti = 1 keV
Slightly dispersive scheme θf = 0.05
Laser I = 1019 W/cm2
x
Conservative scheme θf = 0
ne/nc
1 m3 m
2dx3dv explicit simulation (Calder)• t = 0.05 ω0
-1
2dx3dv implicit simulation• t = 0.141 ω0
-1