Diffusion and accurate hydrodynamics in SPH and grid codes James Wadsley (McMaster) Tom Quinn (Washington), Fabio Governato (Washington), Hugh Couchman (McMaster) Disks 2012, Heidelberg
Dec 18, 2015
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)
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)
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…