18 th International Conference on Numerical Simulation of Plasmas Cape Cod, Massachusetts

1. Asymmetric PML for the Absorption of Waves.Application to Mesh Refinement in Electromagnetic Particle-In-Cell Plasma Simulations. J.-L. Vay Lawrence Berkeley National Laboratory, California, USA J.-C. Adam, A. Héron CPHT, Ecole Polytechnique, France 18th International Conference on Numerical Simulation of Plasmas Cape Cod, Massachusetts September 8, 2003

2. Motivation • Study of laser-plasma interaction in context of fast ignition involves plasma density far greater than critical density • Imposes very strict conditions on mesh size and time steps • Following the system on experimentally realistic space and time scales implies large domains • I.e. boundary conditions are sufficiently remote so that they do not contaminate the physics inside the target • A regular grid results in a lot of wasted resources in modeling large areas of vacuum or low density plasma • Mesh refinement allows finer gridding of localized area but is challenging for electromagnetic PIC: need efficient absorbing mechanism at patch boundary • We present a new Perfectly Matched Layer for the absorption of waves which gives very high absorption rate • A new mesh refinement strategy which takes advantage of this new PML is introduced and tested on a laser-plasma interaction example in the context of the fast ignitor

3. If and => Z=Z0: no reflection. Asymmetric Perfectly Matched Layer (APML) Principle of PML: Field vanishes in layer surrounding domain. Layer medium impedance Z matches vacuum’s Z0 Maxwell Berenger PML Asymmetric PML (APML) The APML introduces some asymmetry in absorption rate. Absorption rates strictly equals for PML and APML at infinitesimal limit. However, absorption rates discretized algorithms differ. If with u=(x,y) => Z=Z0: no reflection.

4. Plane wave analysis PML versus APML Standard PML PML-matched coefficients APML-Hybrid ([3]) APML-LWA ([1])

5. Plane wave analysis PML versus APML for t=2p/w~20dx/c • Best tested APML implementation overall better than best tested PML implementation (for more on this, see [1])

6. R2 A R1 G Mesh refinement • most mesh refinement rely on algorithm ‘sewing’ grids at boundary • an algorithm is applied at the patch boundary to connect the patch and the main grid solution • several solutions have been proposed, using finite-volume, centered finite-difference with ‘jumps’ inside fine grid to get to relevant data, energy conserving schemes, apply different formula depending on direction of wave [2], … • as can be shown on simple 1-D example (see next slide), most produce reflection of waves for wavelengths below coarse grid cutoff, eventually with amplification => instability

7. Tests of various mesh refinement schemes in 1-D o: E, x:B Space only Space+Time (for more on this, see [2])

8. We propose an alternative method by substitution Inside patch: F = F(G)-F(P1)+F(P2) • normal PIC in main grid G at resolution R1 • in area A • patch P1 at res. R1 • patch P2 at res. R2 • both terminated by APML • linear charge deposition on P2 and propagated on P1 and G • when gathering force, force at low resolution R1 is substituted by force at higher resolution R2 on patch P2 R2 Absorbing BCs P2 R1 P1 R1 Outside patch: F = F(G) A G

9. Particle entering and leaving patch • Ideally, the field associated with a macroparticle entering/leaving a patch should (magically) appear/vanish • Since this may be challenging, we have opted for an operationally simple procedure • The current of a macroparticle is deposited inside a patch as soon as it enters it and stops being deposited when it leaves it • This implies the creation of a macroparticle of opposite sign at the entrance location and a macroparticle of same sign at the exit location • With the substitution operation F(G)-F(P1)+F(P2) inside the patch, the contribution due to these standing charge should cancel out • Because this cancellation is not exact (two different resolutions), a residual spurious standing field appears. • Since it is expected that this field will vanish rapidly inside the patch, we define a band on the border of the patch in which we do not perform the substitution

10. The Particle-In-Cell code used for testing: EMI2D • PIC electromagnetic 2D, linear or cubic splines, Esirkepov current deposition scheme (similar to Vilasenor-Buneman algorithm but extend to high-order splines) • Boundary conditions: open system • particles • ions leave the box freely • electrons reflected until an ion exit (overall charge conserved) • EM fields: APML absorbing layer + incoming wave

11. Test: laser interaction with cylindrical target • A laser impinges on a cylindrical target which density is far greater than the critical density (context of fast ignition [4]) • The center of the plasma is artificially cooled to simulate a cold high-density core • Two cases are tested: • Patch boundary in plasma • Patch boundary surrounds plasma Patch: Case 1 Case 2 2s=28/k0 core Laser beam l=1mm, 1020W.cm-2 (Posc/mec~8,83) 10nc, 10keV • The first case is expected to be especially hard on the method since we anticipate that many electrons will cross the patch boundary.

12. Case 1 Case 1 Case 2 Case 2 X-Y particle-density plots for ions and electrons Very similar. See patch boundary in case 1. Very similar. See patch boundary in case 1.

13. Case 1 Case 1 Case 2 Case 2 X-Vx particle plots for ions and electrons Very similar Very similar Background T° higher in case 1

14. Case 1 Case 1 Case 2 Case 2 Y-Vy particle plots for ions and electrons Very similar Very similar Background T° higher in case 1

15. Case 1 Case 1 Case 2 Case 2 Vx-Vy particle plots for ions and electrons Very similar Very similar

16. Case 1 Case 2 Bz main grid In case 2, the electrons see the Laser light from G and its plasma response from P2. They have the same frequency but different wavelengths due to different numerical dispersion on G and P2. This gives a spurious residual low amplitude wave.

17. Case 1 Case 2 Bz patch P1 In case 2, the zone which absorbs the laser light is in the patch. The plasma response to the laser is clearly recognizable.

18. Case 1 Case 2 Bz patch P2 In case 2, the zone which absorbs the laser light is in the patch. The plasma response to the laser is clearly recognizable.

19. Case 1 Case 2 Ex main grid Very similar

20. Case 1 Case 2 Ex Patch P1 In both cases, the accumulation of charge due to macroparticles entering or leaving the effective area of the patch is evident.

21. Case 1 Case 2 Ex Patch P2 In both cases, the accumulation of charge due to macroparticles entering or leaving the effective area of the patch is evident.

22. Case 1 Case 2 Ey main grid In case 2, the electrons see the Laser light from G and its plasma response from P2. They have the same frequency but different wavelengths due to different numerical dispersion on G and P2. This gives a spurious residual low amplitude wave.

23. Case 1 Case 2 Ey Patch P1 In both cases, the accumulation of charge due to macroparticles entering or leaving the effective area of the patch is evident.

24. Case 1 Case 2 Ey patch P2 In both cases, the accumulation of charge due to macroparticles entering or leaving the effective area of the patch is evident.

25. Discussion • The results from the performed test appear very promising since the main features of the physical processes were retained and no instability has been observed. • We note, however, the presence of two spurious effects • when the laser-plasma interaction occurs inside the refined area, different numerical dispersion in the refined patch and the main grid accounts for a spurious, although low intensity, laser trace in the plasma, due to inexact cancellation of the incident laser and the plasma response, • when the patch lies inside the plasma, its boundary is visible as a low density line in the plasma density plots for both species. Several explanations for this effect are being considered: spurious field from remaining charges at boundaries, different cutoffs in plasma frequency on G and P2, a bug,… • Despite these spurious effects, we note that the phase-space projections look very similar, indicating that the macroparticle trajectories were largely unaffected.

26. Conclusion • A New Asymmetric PML was introduced and higher absorption rates were obtained compared with standard PML. • Taking advantage of these high absorption rates, a new strategy for coupling the mesh refinement technique to electromagnetic Particle-In-Cell simulations was devised. • A first test exhibited spurious effects which, nonetheless, did not affect significantly the main physical aspects. • A more profound analysis of the issues will be performed in order to unequivocally identify the source of the spurious effects and remedies will be explored. • Based on our present understanding, these may involve • use of higher-order (less dispersive) Maxwell solver, • add Gauss corrector in patch (Boris, Marder or hyperbolic) to remove standing charges due to macroparticle entering or leaving patch, • devise more elaborate procedure for particle entrance and exit of patch which lead to reduction in magnitude, or even inexistence, of standing charges.

27. References • J.-L. Vay, “Asymmetric Perfectly Matched Layer for the Absorption of Waves” ”, J. Comp. Physics183, 367-399 (2002) • J.-L. Vay, “An extended FDTD scheme for the wave equation. Application to multiscale electromagnetic simulation”, J. Comp. Physics167, 72-98 (2001) • J.-L. Vay, “A new absorbing layer boundary condition for the wave equation”,J. Comp. Physics165, 511-521 (2000) • M. Tabak et al., “Ignition and high gain with ultrapowerful lasers”, P. of Plasmas, Vol. 1, Issue 5, 1626-1634 (1994)