Large scale simulations of astrophysical turbulence Axel Brandenburg (Nordita, Copenhagen) Wolfgang Dobler (Univ. Calgary) Anders Johansen (MPIA, Heidelberg) Antony Mee (Univ. Newcastle) Nils Haugen (NTNU, Trondheim) etc. (...just google for Pencil Code)

2 Overview History: as many versions as there are people?? Example of a cost effective MPI code –Ideal for linux clusters –Pencil formulation (advantages, headaches) –(Radiation: as a 3-step process) How to manage the contributions of 20+ people –Development issues, cvs maintainence Numerical issues –High-order schemes, tests Peculiarities on big linux clusters –Online data processing/visualization

3 Pencil Code Started in Sept with Wolfgang Dobler High order (6 th order in space, 3 rd order in time) Cache & memory efficient MPI, can run PacxMPI (across countries!) Maintained/developed by many people (CVS!) Automatic validation (over night or any time) Max resolution so far , 256 procs

4 Range of applications Isotropic turbulence –MHD (Haugen), passive scalar (Käpylä), cosmic rays (Snod, Mee) Stratified layers –Convection, radiative transport (T. Heinemann) Shearing box –MRI (Haugen), planetesimals, dust (A. Johansen), interstellar (A. Mee) Sphere embedded in box –Fully convective stars (W. Dobler), geodynamo (D. McMillan) Other applications and future plans –Homochirality (models of origins of life, with T. Multamäki) –Spherical coordinates

5 Pencil formulation In CRAY days: worked with full chunks f(nx,ny,nz,nvar) –Now, on SGI, nearly 100% cache misses Instead work with f(nx,nvar), i.e. one nx-pencil No cache misses, negligible work space, just 2N –Can keep all components of derivative tensors Communication before sub-timestep Then evaluate all derivatives, e.g. call curl(f,iA,B) –Vector potential A=f(:,:,:,iAx:iAz), B=B(nx,3)

6 A few headaches All operations must be combined –Curl(curl), max5(smooth(divu)) must be in one go –out-of-pencil exceptions possible rms and max values for monitoring –call max_name(b2,i_bmax,lsqrt=.true.) –call sum_name(b2,i_brms,lsqrt=.true.) Similar routines for toroidal average, etc Online analysis (spectra, slices, vectors)

7 CVS maintained pserver (password protected, port 2301) –non-public (ci/co, 21 people) –public (check-out only, 127 registered users) Set of 15 test problems in the auto-test –Nightly auto-test (different machines, web) Before check-in: run auto-test yourself Mpi and nompi dummy module for single processor machine (or use lammpi on laptops)

8 Switch modules magnetic or nomagnetic (e.g. just hydro) hydro or nohydro (e.g. kinematic dynamo) density or nodensity (burgulence) entropy or noentropy (e.g. isothermal) radiation or noradiation (solar convection, discs) dustvelocity or nodustvelocity (planetesimals) Coagulation, reaction equations Homochirality (reaction-diffusion-advection equations)

9 Features, problems Namelist (can freely introduce new params) Upgrades forgotten on no-modules (auto-test) SGI namelist problem (see pencil FAQs)

10 Pencil Code check-ins

11 High-order schemes Alternative to spectral or compact schemes –Efficiently parallelized, no transpose necessary –No restriction on boundary conditions –Curvilinear coordinates possible (except for singularities) 6th order central differences in space Non-conservative scheme –Allows use of logarithmic density and entropy –Copes well with strong stratification and temperature contrasts

12 (i) High-order spatial schemes Main advantage: low phase errors

13 Wavenumber characteristics

14 Higher order – less viscosity

15 Less viscosity – also in shocks

16 (ii) High-order temporal schemes Main advantage: low amplitude errors 3 rd order 2 nd order 1 st order 2N-RK3 scheme (Williamson 1980)

17 Shock tube test

18 Hydromagnetic turbulence and subgrid scale models? Want to shorten diffusive subrange –Waste of resources Want to prolong inertial range Smagorinsky (LES), hyperviscosity, … –Focus of essential physics (ie inertial range) Reasons to be worried about hyperviscosity –Shallower spectra –Wrong amplitudes of resulting large scale fields

19 Simulations at With hyperdiffusivity Normal diffusivity Biskamp & Müller (2000)

20 The bottleneck: is a physical effect Porter, Pouquet, & Woodward (1998) using PPM, meshpoints Kaneda et al. (2003) on the Earth simulator, meshpoints (dashed: Pencil-Code with ) compensated spectrum

21 Bottleneck effect: 1D vs 3D spectra Compensated spectra (1D vs 3D)

22 Relation to ‘laboratory’ 1D spectra

23 Hyperviscous, Smagorinsky, normal Inertial range unaffected by artificial diffusion Haugen & Brandenburg (PRE, astro-ph/ ) height of bottleneck increased onset of bottleneck at same position

processor run at

25 Structure function exponents agrees with She-Leveque third moment

26 Helical dynamo saturation with hyperdiffusivity for ordinary hyperdiffusion ratio 125 instead of 5

27 Slow-down explained by magnetic helicity conservation molecular value!!

28 MHD equations Induction Equation: Magn. Vector potential Momentum and Continuity eqns

29 Vector potential B=curlA, advantage: divB=0 J=curlB=curl(curlA) =curl2A Not a disadvantage: consider Alfven waves B-formulation A-formulation 2 nd der once is better than 1 st der twice!

30 Comparison of A and B methods

31 Wallclock time versus processor # nearly linear Scaling 100 Mb/s shows limitations Gb/s no limitation

32 Sensitivity to layout on Linux clusters yprox x zproc 4 x 32  1 (speed) 8 x 16  3 times slower 16 x 8  17 times slower Gigabit uplink 100 Mbit link only 24 procs per hub

33 Why this sensitivity to layout? All processors need to communicate with processors outside to group of 24

34 Use exactly 4 columns Only 2 x 4 = 8 processors need to communicate outside the group of 24  optimal use of speed ratio between 100 Mb ethernet switch and 1 Gb uplink

35 Fragmentation over many switches

36 Pre-processed data for animations

37 Ma=3 supersonic turbulence

38 Animation of B vectors

39 Animation of energy spectra Very long run at resolution

40 MRI turbulence MRI = magnetorotational instability w/o hypervisc. t = 600 = 20 orbits w/o hypervisc.  t = 60 = 2 orbits

41 Fully convective star

42 Geodynamo simulation

43 Homochirality: competition of left/right Reaction-diffusion equation

44Conclusions Subgrid scale modeling can be unsafe (some problems) –shallower spectra, longer time scales, different saturation amplitudes (in helical dynamos) High order schemes –Low phase and amplitude errors –Need less viscosity 100 MB link close to bandwidth limit Comparable to and now faster than Origin 2x faster with GB switch 100 MB switches with GB uplink +/- optimal

45 Transfer equation & parallelization Analytic Solution: Ray direction Intrinsic Calculation Processors

46 The Transfer Equation & Parallelization Analytic Solution: Ray direction Communication Processors

47 The Transfer Equation & Parallelization Analytic Solution: Ray direction Processors Intrinsic Calculation

48 Current implementation Plasma composed of H and He Only hydrogen ionization Only H - opacity, calculated analytically No need for look-up tables Ray directions determined by grid geometry No interpolation is needed

49 Convection with radiation