1. Additional Slides VALIS

2. VALIS – intro.  Any kinetic model of plasma will be closely related to Vlasov’s equation  Describes evolution of particle density, in response to self- consistent fields from Maxwell’s equations in a 6D phase space  (3 x space, 3 x momentum)  For now, restrict our selves to two spatial and two momentum dimensions: (x, y, u x, u y ).  A ‘2D2P’ model  N.J. Sircombe and T.D. Arber, J. Comp. Phys. 228, 4773, (2009)

3. Approach  Take the 2D2P phase space and ‘grid’ it up  f is then a 4D fluid  We can build the algorithm on ones developed for Eulerian fluid codes.  Operator splitting. Split updates of 4D phase space into a series of 1D updates, interleaved to ensure the complete timestep is symmetric and time-centred 1.Update in x for ½ a step 2.Update in y for ½ a step 3.Update in u x for ½ a step 4.Update in u y for 1 step 5.Update in u x for ½ a step 6.Update in y for ½ a step 7.Update in x for ½ a step  Each of these updates is then just a 1D advection

4. VALIS Scaling  Scaling  2D2P Vlasov problems can become very large, very fast*  Must make efficient use of HPC Some choices (such as any non-local elements to algorithm) can make this very difficult  Again: the explicit, split, conservative approach pays dividends – it can be parallelised via domain decomposition, across all four dimensions, and scales well.  Cost of each doubling of n pe, is negligible  Parallel IO is also a necessity, and included in VALIS’ IO subsystem Relative increase in runtime vs. n pe on CRAY XT3, triangles represent dual- core nodes, squares single core. * e.g. 2D2P SP-LPI problem with mobile ions (1024, 512, 256, 256) => 512 Gb memory footprint, and therefore >512Gb restart dumps.

5. 1D 500nc problem - T hot estimate  T hot peaks and falls (insufficient dumps to establish if a ‘steady state’ is reached).  Need to repeat runs to correct problems  Some clipping of distribution at ~12 MeV due to the momentum domain being too small  Immobile ions Time (fs)FrontIntegratedRear 150NA 2007.3 MeV5.6 MeVNA 3005.3 MeV6.7 MeV3.5 MeV 4001.5 MeV4.3 MeV5.0 MeV 5005.0 MeV3.0 MeV1.8 MeV

6. Application in 2D (keV) ‘long’ ‘medium’ ‘short’ 0 degrees 10 degrees 30 degrees

7. 2D ‘Long’

8. 2D ‘Medium’

9. 2D ‘Short’

10. VALIS Future Development  Addition of:  Multi-species physics  Mobile ions, ponderomotive steepening etc.  Collisional physics (Krook operator)  Transport in dense material  In anticipation of future, massively parallel HPC Optimisation of:  Domain decomposition scheme  Communications  Core algorithm

11. Additional Slides EPOCH

12. Introduction  Particle-in-cell simulations of the 1D and 2D test problems performed using EPOCH (Extendable Open PIC Collaboration)  Runs in 1D performed with mobile and immobile ions  Problems with self-heating encountered in 2D – flatten ramp at 100n c instead of 500n c

13. 1D test problem  Density profile at t=500fs  Red = electrons Blue = ions Green = electrons with immobile ions  Little deformation of front surface observed with immobile ions

14. 2D test problem  Runs performed with mobile and immobile ions  Electron density profile at t=250ps with mobile (bottom) and immobile (top) ions  Immobile ion run used 80x80 micron box  Mobile ion run used 40x40 micron box (to reduce self- heating)

15. 2D test problem contd.  x-px electron phase space plots at t=100fs (left) and t=250fs (right) for mobile ion case