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)  

Long irradiation simulations : hot electron generation

Explicit relativistic Implicit relativistic

Long irradiation simulations : Quasistatic magnetic field generation

Implicit relativistic

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

Over-critical laser plasma interaction (2/2)

Explicit relativistic (Calder)

Implicit relativistic

Total energies

Electronic (left) and ionic (right) phase spaces (x, px)

Electronic (left) and ionic (right) phase spaces (x, px)

Total energies17h30

>48h