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Diffusion and accurate hydrodynamics in SPH and grid codes

Diffusion and accurate hydrodynamics in SPH and grid codes. James Wadsley (McMaster) Tom Quinn (Washington), Fabio Governato (Washington), Hugh Couchman (McMaster). Disks 2012, Heidelberg. Test. Gasoline ( Wadsley , Stadel & Quinn 2004). Agertz et al 2007

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Diffusion and accurate hydrodynamics in SPH and grid codes

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  1. Diffusion and accurate hydrodynamics in SPH and grid codes James Wadsley (McMaster) Tom Quinn (Washington), Fabio Governato (Washington), Hugh Couchman (McMaster) Disks 2012, Heidelberg

  2. Test Gasoline (Wadsley, Stadel & Quinn 2004) Agertz et al 2007 Code comparison paper from the proto AstroSim conference (2004) Gadget (Springel) FLASH ENZO ART

  3. Test SPH Smoothed Particle Hydrodynamics (Monaghan 1992, Springel & Hernquist 2002) Agertz et al 2007 Code comparison paper from the proto AstroSim conference (2004) PPM Piecewise Parabolic Method (Collela & Woodward 1987)

  4. Basic Result: SPH blobs don’t break up Quantitative measure: Fraction of cloud remaining above 64% of initial density • As cloud fragments • – size R halves every Kelvin-Helmholtz time τKH ~ R/v • Effective Kelvin-Helmholtz time halves repeatedly until cloud catches up with flow (v  0) • SPH: Kelvin-Helmholtz time static

  5. Basic Result: SPH blobs don’t break up Immediate SPH issue: Surface Tension present in arithmetic sum Pressure force (e.g. Monaghan 1992, Gasoline, Gadget, …) Suppresses Kelvin Helmholtz instabilities  Issue first identified by Ritchie & Thomas (2001)

  6. SPH Kelvin Helmholtz fixed • Ritchie & Thomas (2001) – smooth pressure not density and Geometric Density Average in Force: remove surface tension (pressure spike at density jump) • Price (2008) -- smear density jumps • Read, Hayfield & Agertz (2010), Read & Hayfield (2011), Abel (2011), Murante et al (2011) … modified SPH • Solutions typically expensive (more accurate)

  7. Key to alleviating SPH surface tension: • Geometric Density Average in Force (GDForce): Morris (1996), Monaghan (1992) Ritchie and Thomas (2001), Can be derived from a Lagrangian: Monaghan & Rafiee (2012) see also Abel (2011) Merging Cluster Test Geometric Density Force Standard Force

  8. Blob Test in Entropy (T3/2/ρ) Hi-Res ENZO Hi-Res Standard SPH

  9. Blob Test in Entropy (T3/2/ρ) Hi-Res ENZO Hi-Res GDForce SPH

  10. Blob Test in Entropy (T3/2/ρ) ENZO Standard SPH GDForce SPH t = 3.75 τKH t = 2.5 τKH t = 1.25 τKH

  11. Blob Test in Entropy (T3/2/ρ) ENZO Standard SPH Low Entropy Blobs Indestructible! GDForce SPH t = 3.75 τKH t = 2.5 τKH t = 1.25 τKH

  12. The second issue: Entropy mixing Cluster Comparison (Frenk et al 1999) Grid codes have entropy cores, SPH codes don’t (because they don’t mix) ENZO SPH Wadsley, Veeravalli & Couchman (2008)

  13. How to get entropy cores? • Shocks (while cs<v) • Mix hot & cold cluster gas SPH can’t: Eulerian codes can (accidentally):

  14. Subgrid Turbulent Mixing • Fluid elements on a fixed (resolved) physical scale do exchange energy/entropy due to unresolved (turbulent) motions Turbulent diffusive heat flux

  15. Ways to model turbulent diffusion: • Lowest-order turbulent diffusion model: Turbhas units of velocity x length Smagorinksymodel (1963): Assumes Prandtl number ~ 1 Sij = strain tensor of resolved flow, lSSmagorinsky length Incompressible grid models set ls2~ 0.02 x 2 (Lilly 1967) For SPH we can try Turb= C h2 S C ~ 0.1 Wadsley, Veeravalli & Couchman (2008) Shen, Wadsley & Stinson (2010)

  16. Ways to model turbulent diffusion: • Lowest-order turbulent diffusion model: Cluster Entropy Cores easily obtained in SPH with thermal diffusion included – solved in 2008 Wadsley, Veeravalli & Couchman (2008) Shen, Wadsley & Stinson (2010)

  17. Bottom Line on Diffusion & SPH • Physical diffusion in needed in all simulations e.g. Metals should mix in galactic outflows (Shen, Wadsley & Stinson 2010) With thermal diffusion in SPH: • Get entropy cores in Galaxy Clusters • Necessary to model entropy of mixing (Springel 2010) • Better Kelvin-Helmholtz • Better Blob results … (but not quite there) • … Need Geometric Density Force AND Diffusion

  18. Blob Test in Entropy (T3/2/ρ) Hi-Res ENZO GDForce + Turbulent Diffusion SPH

  19. Blob Test in Entropy (T3/2/ρ) ENZO Standard SPH GDForce + Turbulent Diffusion SPH t = 3.75 τKH t = 2.5 τKH t = 1.25 τKH

  20. Is grid (PPM) the right answer?No: Numerical Diffusion Approximate, e.g. is not Galilean Invariant ENZO Moving flow ENZO 1/2 – 1/2 ENZO Moving blob t = 3.75 τKH t = 2.5 τKH t = 1.25 τKH

  21. Blob’s falling apart… Standard SPH Standard SPH + Diffusion GDForce SPH Dense Cloud Remaining GDForce + Diffusion Coefficient 0.1,0.03,0.01 t / τKH

  22. Blob’s falling apart… (Log) Standard SPH Standard SPH + Diffusion GDForce SPH Dense Cloud Remaining GDForce + Diffusion Coefficient 0.1,0.03,0.01 t / τKH

  23. Blob’s falling apart… (Log) Standard SPH Standard SPH + Diffusion GDForce SPH Dense Cloud Remaining ENZO (1/2 each) ENZO Moving blob ENZO Moving flow (Agertz et al.) GDForce + Diffusion Coefficient 0.1,0.03,0.01 t / τKH

  24. Diffusion coefficient… ENZO @ 4.5 Gyr GASOLINE @ 4.5 Gyr Smash galaxy clusters together: Clean version of Frenk et al. ( 1999) with no substructure 500 pc Results look the same but ENZO mixes more in the core (see left) than SPH+GDForce+Diffusion Gasoline diffusion coefficients: Also: Mass metallicity + IGM Metals ENZO Entropy at 20 Gyr Gasoline 1.0 0.1, 0.03, 0.01, no Diffusion Inner 100 kpc

  25. Diffusion coefficient… ENZO @ 4.5 Gyr GASOLINE @ 4.5 Gyr Smash smooth galaxy clusters together: Clean version of Frenk et al. ( 1999) with no substructure 500 pc Results look the same but ENZO mixes more in the core (see left) than SPH+GDForce+Diffusion Gasoline diffusion coefficients: Also: Mass metallicity + IGM Metals ENZO Entropy at 20 Gyr Gasoline 1.0 0.1, 0.03, 0.01, no Diffusion Inner 100 kpc

  26. Galaxy Formation:Kaufmann et al (2008) blobs no more Standard SPH GDForce+Diffusion SPH Toy Galaxy Model (cf. Kaufmann et al.), 20 kpc wide edge on view Conclusion: Blobs product of SPH surface tension effects (see also: Joung et al 2011)

  27. Galaxy Formation:Smoother disks & spiral structure Standard SPH GDForce+Diffusion SPH Toy Galaxy Model (cf. Kaufmann et al.), ~ 12 kpc wide face on view Smoother Structure

  28. Galaxy Formation Standard SPH GDForce+Diffusion 1011 Solar Mass Galaxy, z=0.6 (latest output) Initial indications: Cold flow mass only ~ 2% different Increased star formation. Need to understand how marginally resolved cooling works

  29. Conclusions • Geometric Density Average Force alleviates surface tension affects in SPH • Well-characterized diffusion treatments necessary for all codes (SPH & Grid) • Careful attention to resolution scale behaviour important (e.g. Instabilities: KH, Jeans, Thermal/Cooling…) • Gasoline public release: including corrected force (blob free!), Stinson et al. star formation and feedback, cooling, planetesimal collisions – coming to google code this summer…

  30. Sedov Test GDForce

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