Computational Astrophysics R Capuzzo Dolcetta Legnaro 16 febbraio

Computational Astrophysics R. Capuzzo Dolcetta Legnaro, 16 febbraio 2011

Computational astrophysics… Computational Physics and Astrophysics • Physics is characterized by mathematical models apt to describe physical systems behaviours. • Anyway, the ab initio approach to the model equations is not always possible, especially in the case of QM where solutions in closed form are available only for a bunch of simple models. • Consequently, resorting to numerical approximations is needed

Computational astrophysics… Computational Physics Main fields: • quantum mechanics: LQCD, 4 D non perturbative approach • molecular dynamics: proteines biomolecules, materials science, complex systems • fluid dynamics: transonic flows , turbulence, meteorology • magneto fluid dynamics: Navier-Stokes + Maxwell. magnetic flux conservation Particular problems are due to

Computational astrophysics… Long-range forces • Unscreened Coulomb systems • Ion beams • Wave-particle interactions in collisionless plasmas • Self-gravitating systems

Computational astrophysics… Present Grand Challenges in computational astrophysics: ● ● ● How do Stars form? (gravitational radiative hydrodynamics at very large R) Why and how Supernova explode? (relativistic radiative hydro) How do massive black holes form and violently emit EM and G waves? (relativistic mass accretion onto a massive black hole; binary black holes merging) How can we understand the Large Scale Structure of the Universe? (very many-body, low prec. problem) Is the Solar system stable? (few-body, very high prec. Problem)

Computational astrophysics… From Nature (2009, 459, 817): “Existence of collisional trajectories of Mercury, Mars and Venus with the Earth”, Laskar & Gastineau “…In a set of 2, 501 orbits with i. c. that are in agreement with our present knowledge of the parameters of the Solar System, we found, as in previous studies, that 1% of the solutions lead to a large increase in Mercury’s eccentricity –an icrease large enough to allow collisions with Venus or the Sun. More surprisingly, in one of these high-eccentricity solutions, a subsequent decrease in Mercury’s ecc. induces a transfer of angular momentum from the giant planets that destabilizes all the terrestrial planets ≈3. 34 Gyr from now, with possible collisions of Mercury, Mars or Venus with the Earth. ”

Computational astrophysics… Peculiarity of astrophysical simulations is the role of self-gravity self grav/ext grav 50 km 30 pc = 90 ly = 6 x 106 AU 1 Mpc = 30 Mly = 2 GAU Garda lake GC: M 13 Galaxy cluster ~10 -8 ~10 -2

Computational astrophysics… Self-gravitating systems are difficult to study because Uij 1/rij is double-divergent 1) UV divergence ( ) → Δt → 0 2) IR divergence (Uij never vanishes) O(N 2) Multiple space-time scale problem Cannot resort to perturbative methods

Computational astrophysics… Classic gravitational N-body problem • 10 integrals of motion, independently of N; (Poincarè and Bruns show that is impossible finding other integrals as algebraic functions of ri and/or vi) Analytical solution for N=2 and by series for N=3 and L>0 (… Poincarè won the king Oscar prize. . . )

Computational astrophysics… The most expensive part is the euclidean distance computation… It requires solution of a non-linear equation: although NR iteration converges quadratically the single pair force evaluation Fij requires 25 flops…

Computational astrophysics… Force Fij evaluation : 25 flop nf= # of op. per time step with a PE v=10 Gflop/s, single pair (N=2) tij =2. 5 10 -9 sec N=1000 nf =1. 25 107 flop t = 1. 25/1000 sec N=105 nf =1. 25 1011 flop t = 12. 5 sec N=1011 nf =1. 25 1023 flop t = 1. 25 1013 sec ≈ 4 105 yr!

Computational astrophysics… What if Newton had been more “friendly” with a 1/rij 3 force? A factor ≥ 6 gain in FP operations per every pair and contribution of distant objects would scale logarithmically All this for holds for “dry” systems (pure N bodies)

Computational astrophysics… Real astrophysical systems are not simple N-bodies. . . a condensed phase ( s) in a dilute medium (g) continuity eq. g gas motion eq. g+ energy eq. g stellar motion eq. g+ Poisson’s eq. g eq. of state g pressure force p (short-range) gravity force U (long-range)

Computational astrophysics… 3 D self-gravitating astrophysical systems may be suitably simulated in a Lagrangian way (particle systems: =N bodies, g=SPH) …nevertheless. . . small scale fluctuations of p(r) introduce large fluctuations of p Low computational cost paid by low precision the body force requires (NSPH+N*)2 valutations High computational cost high precision

Computational astrophysics… Profiling in a typical simulation Task Cpu time (%) Gravitational forces eval. 60 Small scale forces (fluid-dynamic forces) 25 Time integration 15

Computational astrophysics… How attack the problem? ▪Via solution of Poisson’s eq. on a grid (FFT), multipolar expansions, tree-algorithms, in general PM and PP-PM techniques ▪ By mean of supercomputers

Computational astrophysics… Computers: multi-purpose, specializzati e dedicati. Multipurpose Super. Computers: TIER 0 Blue Gene/P in Juelich (DE 1 PFlop/s Bull “Curie” in TGCC (FRA) , 1. 6 PFlop/s in LQCD: APE Specialized and/or dedicated in astrophysics: GRAPE-6: 64 Tflops

Computational astrophysics… For certain categories of problems, seeking for both high precision and high velocity a valid alternative is using local hybrid architectures.

Computational astrophysics… A cheap approach to HPC: Graphic Processing Units TESLA C 2050 - 2070 448 cores, 3 -6 GB memory, 1. 15 GHz per core. , 144 GB/s mem. b. width, 1 TFlop/s AMD Radeon HD 6970 1536 cores, 2 GB memory, 880 MHz per core, 144 GB/s mem. b. width, 2. 7 1 TFlop/s in SP, 683 Gflop/s in DP

Computational astrophysics… + = Power: ~ 12 Gflops (CPU) ~ 2 Tflops (GPU) 2 exacore Xeon 2 Ghz clock 2 TESLA C 2050 Cost: ~ 7500 euro ~ 1100 W

Computational astrophysics… NBSymple (Capuzzo Dolcetta, Mastrobuono Battisti) an N-body, time-symplectic code for composite CPU+GPU computing systems astrowww. phys. uniroma 1. it/dolcetta/NBSymple. html Symplectic algorithms: what are they? Advantages Time reversibility Volume preserving in the phase-space: excellent for collisionless systems Good secular energy conservation; non-sympl. integrators are dissipative Disadvantages Computationally demanding