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

Agertz et al 2007Code comparison

paper from the proto AstroSim conference (2004)

Gasoline(Wadsley,

Stadel & Quinn 2004)

Gadget(Springel)

FLASH

ENZO

ART

Test

Agertz et al 2007Code comparison

paper from the proto AstroSim conference (2004)

PPMPiecewiseParabolic

Method(Collela &

Woodward 1987)

SPHSmoothed

ParticleHydrodynamics

(Monaghan 1992, Springel & Hernquist 2002)

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

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 byRitchie & Thomas (2001)

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)

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)

ababa

ba

bb

a WPP

mdt

dv

Standard Force Geometric Density Force

Merging Cluster Test

Blob Test in Entropy (T3/2/ρ)

Hi-Res ENZO

Hi-Res Standard SPH

Blob Test in Entropy (T3/2/ρ)

Hi-Res ENZO

Hi-Res GDForce SPH

Blob Test in Entropy (T3/2/ρ)

ENZO

GDForceSPH

StandardSPH

t = 1.25 τKH t = 3.75 τKH t = 2.5 τKH

Blob Test in Entropy (T3/2/ρ)

ENZO

GDForce SPH

StandardSPH

t = 1.25 τKH t = 3.75 τKH t = 2.5 τKH

Low Entropy Blobs Indestructible!

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)

How to get entropy cores?

• Shocks (while cs<v)• Mix hot & cold cluster gas

SPH can’t:

Eulerian codes can (accidentally):

flowfollowing

constPsAu

dt

du

.)()v.( )1(

)v.( )1(.v

uerrors

advectionu

t

u

Subgrid Turbulent Mixing

• Fluid elements on a fixed (resolved) physical scale do exchange energy/entropy due to unresolved (turbulent) motions

0 part, unresolved

, field ofpart (filtered) resolved where

v).()v.( )1(.v.v

togoes )v.( )1(.v

aa

aa

uuuut

u

uut

u

Turbulent diffusive heat flux

Ways to model turbulent diffusion:

• Lowest-order turbulent diffusion model:

Turb has units of velocity x lengthSmagorinksy model (1963):

Assumes Prandtl number ~ 1 Sij = strain tensor of resolved flow, lS Smagorinsky lengthIncompressible grid models set ls

2 ~ 0.02 x 2 (Lilly 1967)

For SPH we can try Turb = C h2 S C ~ 0.1

ijijSTurb SSSSl ,2

uuut

u

Turb )v.()1(.v

Wadsley, Veeravalli & Couchman (2008) Shen, Wadsley & Stinson (2010)

Ways to model turbulent diffusion:

• Lowest-order turbulent diffusion model:

uuut

u

Turb )v.()1(.v

Wadsley, Veeravalli & Couchman (2008) Shen, Wadsley & Stinson (2010)

Cluster Entropy Cores easily obtained in SPH with thermal diffusion included – solved in 2008

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

Blob Test in Entropy (T3/2/ρ)

Hi-Res ENZO

GDForce + Turbulent Diffusion SPH

Blob Test in Entropy (T3/2/ρ)

ENZO

GDForce +TurbulentDiffusionSPH

StandardSPH

t = 1.25 τKH t = 3.75 τKH t = 2.5 τKH

Is grid (PPM) the right answer?No: Numerical Diffusion Approximate,

e.g. is not Galilean Invariant

ENZOMoving flow

t = 1.25 τKH t = 3.75 τKH t = 2.5 τKH

ENZO1/2 – 1/2

ENZOMovingblob

Blob’s falling apart…

t / τKH

Den

se C

loud

Re

mai

ning

GDForce + Diffusion Coefficient 0.1,0.03,0.01

Standard SPHStandard SPH + DiffusionGDForce SPH

Blob’s falling apart… (Log)

t / τKH

Den

se C

loud

Re

mai

ning

GDForce + Diffusion Coefficient 0.1,0.03,0.01

Standard SPHStandard SPH + DiffusionGDForce SPH

Blob’s falling apart… (Log)

t / τKH

GDForce + Diffusion Coefficient 0.1,0.03,0.01

Standard SPHStandard SPH + DiffusionGDForce SPHENZO (1/2 each)ENZO Moving blobENZO Moving flow (Agertz et al.)D

ense

Clo

ud

Rem

aini

ng

Diffusion coefficient…Smash galaxy clusters together: Clean version of Frenk et al. ( 1999) with no substructure

Results look the same but ENZO mixes more in the core (see left) than SPH+GDForce+DiffusionGasoline diffusion coefficients:

Also: Mass metallicity + IGM Metals Inner 100 kpc

ENZO @ 4.5 Gyr

GASOLINE @ 4.5 Gyr

ENZO

1.0 0.1, 0.03, 0.01, no Diffusion

500 pc

Gasoline

Entr

opy

at 2

0 G

yr

Diffusion coefficient…Smash smooth galaxy clusters together: Clean version of Frenk et al. ( 1999) with no substructure

Results look the same but ENZO mixes more in the core (see left) than SPH+GDForce+DiffusionGasoline diffusion coefficients:

Also: Mass metallicity + IGM Metals Inner 100 kpc

ENZO @ 4.5 Gyr

GASOLINE @ 4.5 Gyr

ENZO

1.0 0.1, 0.03, 0.01, no Diffusion

500 pc

Gasoline

Entr

opy

at 2

0 G

yr

Galaxy Formation:Kaufmann et al (2008) blobs no more

Toy Galaxy Model (cf. Kaufmann et al.), 20 kpc wide edge on viewConclusion: Blobs product of SPH surface tension effects(see also: Joung et al 2011)

Standard SPH GDForce+Diffusion SPH

Galaxy Formation:Smoother disks & spiral structure

Toy Galaxy Model (cf. Kaufmann et al.), ~ 12 kpc wide face on viewSmoother Structure

Standard SPH GDForce+Diffusion SPH

Galaxy Formation

1011 Solar Mass Galaxy, z=0.6 (latest output)Initial indications: Cold flow mass only ~ 2% differentIncreased star formation. Need to understand how marginally resolved cooling works

Standard SPH GDForce+Diffusion

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…

Sedov Test GDForce