Galaxy formation in SPHS Justin Read ETH Zrich | University of - - PowerPoint PPT Presentation

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Galaxy formation in SPHS Justin Read ETH Zrich | University of - - PowerPoint PPT Presentation

Jan 02, 2023 275 likes •708 views

Galaxy formation in SPHS Justin Read ETH Zrich | University of Leicester With: T. Hayfield, A. Hobbs, C. Power Background | Classic SPH N Integral i = m j W ij ( | r ij | , h i ) ( Continuity j N d v i m j dt = i

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Justin Read ETH Zürich | University of Leicester With: T. Hayfield, A. Hobbs, C. Power

Galaxy formation in SPHS

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Background | ‘Classic’ SPH

Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011

Integral Continuity Momentum

  • Eqn. of state

ρi =

N

⇧

j

mjWij(|rij|, hi) (

dvi dt =

N

⇧

j

mj ρiρj (Pi + Pj) ⇤iW ij

⇧

Pi = Aiργ

i

; Ai = const.

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SLIDE 3

dvi dt =

N

X

j

mj Pi ρ2

i

+ Pj ρ2

j

! riW ij

Integral Continuity Momentum

⇧

Pi = Aiργ

i

; Ai = const.

  • Eqn. of state

ρi =

N

⇧

j

mjWij(|rij|, hi) (

i j

Background | The Euler equations (Lagrangian ‘entropy’ form)

Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011

Integral Continuity Momentum

⇧

Pi = Aiργ

i

; Ai = const.

  • Eqn. of state

ρi =

N

⇧

j

mjWij(|rij|, hi) (

‘classic’ SPH

[inc. ‘energy’ form and similar]

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SLIDE 4

Integral Continuity Momentum

⇧

Pi = Aiργ

i

; Ai = const.

  • Eqn. of state

ρi =

N

⇧

j

mjWij(|rij|, hi) (

dvi dt =

N

⇧

j

mj ρiρj (Pi + Pj) ⇤iW ij

i j

Background | The Euler equations (Lagrangian ‘entropy’ form)

Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011

Integral Continuity Momentum

⇧

Pi = Aiργ

i

; Ai = const.

  • Eqn. of state

ρi =

N

⇧

j

mjWij(|rij|, hi) (

dvi dt =

N

⇧

j

mj ρiρj (Pi + Pj) ⇤iW ij

Improved force error

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SLIDE 5

Background | Advantages of SPH

Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011

  • 1. Lagrangian
  • 2. Galilean invariant
  • 3. Manifestly conservative
  • 4. Easy to implement
  • 5. Couples to O(N) FMM gravity
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SLIDE 6

A 1:10 density ratio gas sphere in a wind tunnel (Mach 2.7), initially in pressure eq.

The “blob test”

Background | Some problems with ‘classic’ SPH

Agertz et al. 2007

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SLIDE 7

Background | Some problems with ‘classic’ SPH

Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011

  • 1. The ‘E0’ error
  • 2. Multivalued pressures
  • 3. Overly viscous
  • 4. Noisy
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SLIDE 8
  • 1. The ‘E0 error’ | Taylor expanding the momentum equation

Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011

Pj Pi + hxij · ∇Pi + O(h2)

Momentum

dvi dt =

N

⇧

j

mj ρiρj (Pi + Pj) ⇤iW ij

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SLIDE 9

Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011

Momentum

dvi dt =

N

⇧

j

mj ρiρj (Pi + Pj) ⇤iW ij

dvi dt ⇥ Pi hiρi 2

N

⇧

j

mj ρj ⇤x

i W ij Mi⇤iPi

ρi + O(h) (

⇒Euler eqn. E0

  • 1. The ‘E0 error’ | Taylor expanding the momentum equation
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SLIDE 10
  • 1. The ‘E0 error’ | Minimising E0 - raising the kernel sampling

E0 = 2

N

⇧

j

mj ρj ⇤x

i W ij ⇥ 2

⌃

V

dV ⇤xW

x x x xx x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x x x x x x

  • 1. Smooth on kernel scale (stable kernel)
  • 2. Larger neighbour number
  • 3. More power in kernel wings

Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011

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Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011

  • 1. The ‘E0 error’ | Minimising E0 - raising the kernel sampling

CS CT HOCT

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SLIDE 12

`

SPH-CS128

Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011

  • 1. The ‘E0 error’ | Minimising E0 - raising the kernel sampling
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SLIDE 13

SPH-CS128

Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011

  • 1. The ‘E0 error’ | Minimising E0 - raising the kernel sampling

−40 −20 20 40 20 40 60 80 100 x y

SPH-CS128

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SLIDE 14

SPH-CS128

Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011

  • 1. The ‘E0 error’ | Minimising E0 - raising the kernel sampling

−40 −20 20 40 20 40 60 80 100 x y

SPH-CT128

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SLIDE 15

SPH-CS128

Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011

  • 1. The ‘E0 error’ | Minimising E0 - raising the kernel sampling

SPH-HOCT442

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SPH-CS128

Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011

  • 1. The ‘E0 error’ | Minimising E0 - raising the kernel sampling

SPH-HOCT442

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SLIDE 17

Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011

x x x x x x x x x x x x

  • P1 = A1ργ

1

P2 = A2ργ

2

P1 = P2

  • 2. Multivalued pressures | The problem
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P2 = A2ργ P1 = A1ργ

x x x x x x x x x x x x

  • P1 = P2

Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011

  • 2. Multivalued pressures | The problem
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SLIDE 19

20 40 60 80 100 0.8 0.9 1.0 1.1 1.2 1.3 y P P0

SPH-HOCT442

Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011

  • 2. Multivalued pressures | The problem
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SLIDE 20

P A1, m1, v1... A2, m2, v2...

Momentum : Artificial viscosity Entropy : Artificial thermal conductivity Mass : (i.e. for multimass applications)

Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011; Ritchie & Thomas 2001; Price 2008; Wadsley et al. 2008; Cullen & Dehnen 2010

  • 2. Multivalued pressures | An ‘early warning’ switch

Add conservative dissipation:

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Read, Hayfield & Agertz 2010 (RHA10); Read & Hayfield 2011; Ritchie & Thomas 2001; Price 2008; Wadsley et al. 2008; Cullen & Dehnen 2010

  • 2. Multivalued pressures | An ‘early warning’ switch

αloc,i =

⇧

h2

i |⌅(⌅·vi)|

h2

i |⌅(⌅·vi)|+hi|⌅·vi|+nscs αmax

⌅ · vi < 0

  • therwise

[Requires high order gradient estimator] [i.e. going to converge] [i.e. converging]

αi = αloc,i αi < αloc,i

  • therwise, αi smoothly decays back to zero:

˙ αi = (αloc,i αi)/τi αmin < αloc,i < αi ˙ αi = (αmin αi)/τi αmin > αloc,i

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SPHS | Putting it all together

  • 1. ‘E0’ error reduced using 442 neighbours and stable

higher order HOCT kernel. Also much lower noise (4).

  • 2. Multivalued pressures eliminated using advance warning

high order switch and conservative dissipation. Lower viscosity away from shocks (3); multimass particles now possible.

  • 3. Timestep limiter [Saitoh & Makino 2009] => strong

shocks correctly tracked.

  • 4. Implementations in GADGET2 & 3.

Read & Hayfield 2011

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`

Read & Hayfield 2011

SPHS tests | Sedov-Taylor blast wave

SPHS-442

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Read & Hayfield 2011

SPHS tests | Sedov-Taylor blast wave

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Read & Hayfield 2011

SPHS tests | Sedov-Taylor blast wave

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Read & Hayfield 2011

SPHS tests | Sedov-Taylor blast wave

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SLIDE 27

Read & Hayfield 2011

SPHS tests | Gresho vortex

SPHS-442

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SLIDE 28

Read & Hayfield 2011

SPHS tests | Gresho vortex

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SLIDE 29

`

SPHS tests | KH instability 1:8 density contrast ... multimass

Read & Hayfield 2011

SPHS-442 multimass

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SLIDE 30

`

Read & Hayfield 2011

SPHS tests | Blob test

SPHS-442

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Power, Read & Hobbs in prep. 2012

SPHS tests | Santa Barbara test

x8 SPH-32 SPHS-442

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SPHS tests | Santa Barbara test

Power, Read & Hobbs in prep. 2012

SPH-32 SPHS-442 x128

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SPHS tests | Santa Barbara test

Power, Read & Hobbs in prep. 2012

SPH-32 SPHS-442 x128

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Power, Read & Hobbs in prep. 2012

SPHS tests | Santa Barbara test

z = 0 SPH-32 SPHS-442

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Power, Read & Hobbs in prep. 2012

SPHS tests | Santa Barbara test

z = 1 SPHS-442 SPH-32

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SPHS | Cooling halos

SPHS-442 SPH-96

Hobbs, Read & Cole 2012; http://arxiv.org/abs/1207.3814

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Hobbs, Read & Cole 2012; http://arxiv.org/abs/1207.3814

10 5

  • 5
  • 10

kpc

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Hobbs, Read & Cole 2012; http://arxiv.org/abs/1207.3814

  • 1.0
  • 0.5

0.0 0.5 1.0 Jz/Jc 200 400 600 800 1000 Nstars

  • 1.0
  • 0.5

0.0 0.5 1.0 Jz/Jc 200 400 600 800 1000 Ngas

Stars Gas SPH SPHS

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SLIDE 39

Hobbs, Read & Cole 2012; http://arxiv.org/abs/1207.3814

SPH SPHS Density Density Pressure Pressure

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Hobbs, Read & Cole 2012; http://arxiv.org/abs/1207.3814

SPH SPHS Pressure Pressure Entropy Entropy

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SPHS-442 | 5M

Hobbs, Read & Cole 2012; http://arxiv.org/abs/1207.3814

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SPHS | Conclusions

  • ‘E0’ error reduced using 442 neighbours and stable

higher order HOCT kernel. Much lower noise.

  • Multivalued pressures eliminated using advance warning

high order switch and conservative dissipation. Lower viscosity away from shocks; multimass particles now possible.

  • Timestep limiter => strong shocks correctly tracked.
  • Good performance and convergence to >1% accuracy
  • n a wide range of test problems.
  • Santa Barbara test => entropy profile core
  • Cooling halos => no SPH blobs